A real function f is said to be continuous at x = c, where c is any point in the domain of f if

Since, h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

 …….equation 1

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain

Function is changing its nature (or expression) at x = 0, so we need to check its continuity at x = 0 first.

f (0) = 0 [using equation 1]

LHL ≠ RHL ≠ f (0)

∴ Function is discontinuous at x = 0

c be any real number such that c > 0

=>  f (x) is continuous everywhere for x > 0.

Let c be any real number such that c < 0

Therefore f (c) =

=> f (x) is continuous everywhere for x < 0.

Hence, f (x) is continuous for all Real numbers except zero that is discontinuous at x = 0.