Find all points of discontinuity of f, where f is defined by f(x)={(|x|/x, if x ≠ 0) (0, if x = 0)

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)={(|x|/x, if x ≠ 0) (0, if x = 0)

It is known that,

x < 0 ⇒ |x| = −x and x > 0 ⇒ |x| = x

Therefore, the given function can be rewritten as

f(x) = {(|x|/x = −x/x = −1, if x < 0) (0, if x=0)

(|x|/x = x/x = 1, if x > 0) 

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) = −1

lim_x→c f(x) = lim_x→c (−1) = −1

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x < 0.

Case II:

If c = 0, then the left hand limit of ff at x = 0 is,

lim_x→0⁻ f(x) = lim_x→0⁻ (−1) = −1

The right hand limit of ff at x = 0 is,

lim_x→0⁺ f(x) = lim_x→0⁺ (1) = 1

It is observed that the left and right hand limit of f at x = 0 do not coincide.

Therefore, f is not continuous at x = 0.

Case III:

If c > 0, then f(c) = 1

lim_x→c f(x) = lim_x→c (1) = 1

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x > 0.

Hence, x = 0 is the only point of discontinuity of f

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 8

Find all points of discontinuity of f, where f is defined by f(x)={(|x|/x, if x ≠ 0) (0, if x = 0)

Summary:

For the function f defined by  f(x)={(|x|/x, if x ≠ 0) (0, if x = 0), f is continuous at all points x, such that x > 0, x = 0 is the only point of discontinuity of f