Find all points of discontinuity of f, where f is defined by

Question

Philoid · Accepted Answer

The given function is

We know that if x > 0

⇒ |x| = -x and

x > 0

⇒ |x| = x

So, we can rewrite the given function as:

The function f is defined at all points of the real line.

Let k be the point on a real line.

Then, we have 3 cases i.e., k < 0, or k = 0 or k >0.

Now, Case I: k < 0

Then, f(k) = -1

= -1= f(k)

Thus,

Hence, f is continuous at all real number less than 0.

Case II: k = 0

= -1

= 1

Hence, f is not continuous at x = 0.

Case III: k > 0

Then, f(k) = 1

= 1 = f(k)

Thus,

Hence, f is continuous at all real number greater than 1.

Therefore, x = 0 is the only point of discontinuity of f.