Continuity
1. Draw the graph of the function f(x) and discuss the continuity of the function f(x) .
(i) f(x) = x+2, when x ≥ 1,
= 3x , when x< 1.
(ii) f(x) = x+1 when x ≤ 2
= 2−x when x > 2 .
(iii) f(x) = |𝑥|
𝑥 ,
(iv) f(x) = |𝑥 − 1|.
2. A function f (x) is defined as f (x) = 5x – 1, for x < 1,
= 4x, for x ≥ 1.
Discuss the continuity at x = 1.
3. A function f (x) is defined as f (x) = x – 2, for x < 3,
= 1 for x = 3,
= 4−x, for x > 3.
Discuss the continuity at x = 3.
4. Discus the continuity of the following functions at the given points:-
(a).
5
25
)( 2
x
x
xf
when x > 5
=10 when x = 5
=
3
153x
when x < 5, at the point x = 5.
(b). f (x) = x2 when x > 2
=5 when x = 2
=2x when x < 2, at the point x=2.
(c). f (x) = 1 + x2 if 0 ≤ x ≤1
=2 – x if x > 1 at the point x = 1.
5. A function
)(x
is defined as follows:-
2
)( xx
for x < 1
= 2.5 for x = 1
= x2 + 2 for x > 1
Is
)(x
continuous at x = 1?
6. Discuss the continuity of the function at x = 4.
f(x) = 𝑥2−16
𝑥−4 , when x ≠ 4,
= 8 , when x= 4.
7. Find the value of k for which the following function is continuous at x=3.
f (x) =
3
9
2
x
x
when x ≠ 3
= k when x = 3.
