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1 | 1918-1921 | (ii)
As a special case of (4) above, if f is the constant function f (x) = λ, then the
function g
λ defined by
( )
( )
gx
g x
λ
λ
=
is also continuous wherever g(x) ≠ 0 In
particular, the continuity of g implies continuity of 1
g The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not |
1 | 1919-1922 | In
particular, the continuity of g implies continuity of 1
g The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous |
1 | 1920-1923 | The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions |
1 | 1921-1924 | They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero |
1 | 1922-1925 | The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 |
1 | 1923-1926 | Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function |
1 | 1924-1927 | The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 |
1 | 1925-1928 | Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number |
1 | 1926-1929 | Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real
number |
1 | 1927-1930 | Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real
number Put x = c + h |
1 | 1928-1931 | Now, observe that f (x) = sin x is defined for every real number Let c be a real
number Put x = c + h If x → c we know that h → 0 |
1 | 1929-1932 | Let c be a real
number Put x = c + h If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function |
1 | 1930-1933 | Put x = c + h If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function |
1 | 1931-1934 | If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function |
1 | 1932-1935 | Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x |
1 | 1933-1936 | Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i |
1 | 1934-1937 | Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i e |
1 | 1935-1938 | Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i e , x ≠ (2n +1) 2
π |
1 | 1936-1939 | This is defined for all real numbers such
that cos x ≠ 0, i e , x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous |
1 | 1937-1940 | e , x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined |
1 | 1938-1941 | , x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions |
1 | 1939-1942 | We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f |
1 | 1940-1943 | Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions |
1 | 1941-1944 | An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c |
1 | 1942-1945 | Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c |
1 | 1943-1946 | The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem |
1 | 1944-1947 | Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function |
1 | 1945-1948 | If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number |
1 | 1946-1949 | The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 |
1 | 1947-1950 | Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function |
1 | 1948-1951 | Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function |
1 | 1949-1952 | The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x |
1 | 1950-1953 | Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function |
1 | 1951-1954 | Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous |
1 | 1952-1955 | Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous |
1 | 1953-1956 | Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 |
1 | 1954-1957 | Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1 |
1 | 1955-1958 | But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 |
1 | 1956-1959 | Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2 |
1 | 1957-1960 | 1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 |
1 | 1958-1961 | Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3 |
1 | 1959-1962 | 2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3 Examine the following functions for continuity |
1 | 1960-1963 | Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3 Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 |
1 | 1961-1964 | 3 Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer |
1 | 1962-1965 | Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5 |
1 | 1963-1966 | (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 |
1 | 1964-1967 | Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1 |
1 | 1965-1968 | 5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1 At x = 2 |
1 | 1966-1969 | Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by
6 |
1 | 1967-1970 | At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
1 | 1968-1971 | At x = 2 Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
1 | 1969-1972 | Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 |
1 | 1970-1973 | 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 |
1 | 1971-1974 | |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 |
1 | 1972-1975 | |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 |
1 | 1973-1976 | , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 |
1 | 1974-1977 | 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function |
1 | 1975-1978 | 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 |
1 | 1976-1979 | 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 |
1 | 1977-1980 | Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 |
1 | 1978-1981 | Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 |
1 | 1979-1982 | 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 |
1 | 1980-1983 | 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18 |
1 | 1981-1984 | 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 |
1 | 1982-1985 | Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1 |
1 | 1983-1986 | 18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1 19 |
1 | 1984-1987 | For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points |
1 | 1985-1988 | What about continuity at x = 1 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x |
1 | 1986-1989 | 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x 20 |
1 | 1987-1990 | Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π |
1 | 1988-1991 | Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21 |
1 | 1989-1992 | 20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x |
1 | 1990-1993 | Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22 |
1 | 1991-1994 | 21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions |
1 | 1992-1995 | Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 |
1 | 1993-1996 | cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 |
1 | 1994-1997 | Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function |
1 | 1995-1998 | 23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25 |
1 | 1996-1999 | Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 |
1 | 1997-2000 | Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26 |
1 | 1998-2001 | Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 |
1 | 1999-2002 | Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 |
1 | 2000-2003 | 26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 |
1 | 2001-2004 | cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 |
1 | 2002-2005 | 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function |
1 | 2003-2006 | 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31 |
1 | 2004-2007 | 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function |
1 | 2005-2008 | Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function 32 |
1 | 2006-2009 | 31 Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function |
1 | 2007-2010 | Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function 33 |
1 | 2008-2011 | 32 Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function |
1 | 2009-2012 | Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function 34 |
1 | 2010-2013 | 33 Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | |
1 | 2011-2014 | Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5 |
1 | 2012-2015 | 34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5 3 |
1 | 2013-2016 | Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5 3 Differentiability
Recall the following facts from previous class |
1 | 2014-2017 | 5 3 Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain |
1 | 2015-2018 | 3 Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists |
1 | 2016-2019 | Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists Derivative of f at c is denoted by f ′(c) or
( ( )) |c
d
dxf x |
1 | 2017-2020 | We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists Derivative of f at c is denoted by f ′(c) or
( ( )) |c
d
dxf x The
function defined by
0
(
)
( )
( )
lim
h
f x
h
f x
f
x
h
→
+
−
′
=
wherever the limit exists is defined to be the derivative of f |
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