Discuss the continuity of the function f , where f is defined by f(x)= {(2x, if x < 0) (0, if 0 ≤ x ≤ 1) (4x, if x > 1)
Solution:
A function is said to be continuous when the graph of the function is a single unbroken curve.
The given function is
f(x)= {(2x, if x < 0) (0, if 0 ≤ x ≤ 1) (4x, if x > 1)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
If c < 0, then f(c) = 2c
limx→c f(x) = limx→c (2x)
= 2c
⇒ limx→c f(x) = f(c)
Therefore, f is continuous at all points x, such that x < 0.
Case II:
If c = 0, then f(0) = 0
The left-hand limit of f at x = 0 is,
limx→0− f(x) = limx→0− (2x)
= 2(0) = 0
The right hand limit of f at x = 0 is,
⇒ limx→0+ f(x) = limx→0+ (0) = 0
Therefore, f is not continuous at x = 0.
Case III:
If 0 < c < 1, then f(x) = 0
limx→c f(x) = limx→c (0) = 0
⇒ limx→c f(x) = f(c)
Therefore, f is continuous at in the interval (0,1).
Case IV:
If c = 1, then f(1) = 0
The left-hand limit of f at x = 1 is,
limx→1− f(x) = limx→1− (0) = 0
The right-hand limit of f at x = 1 is,
limx→1+ f(x) = limx→1+ (4x) = 4(1) = 4
It is observed that the left and right-hand limits of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1.
Case V:
If c < 1, then f(c) = 4c
limx→c f(x) = limx→c(4x)
= 4c
⇒ limx→c f(x) = f(c)
Therefore, f is continuous at all points x such that x > 1.
Hence, f is not continuous only at x = 1
NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 15
Discuss the continuity of the function f , where f is defined by f(x)= {(2x, if x < 0) (0, if 0 ≤ x ≤ 1) (4x, if x > 1)
Summary:
For the function defined by f(x)= {(2x, if x < 0) (0, if 0 ≤ x ≤ 1) (4x, if x > 1), f is continuous at all points x such that x > 1. Hence, f is not continuous only at x = 1
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