The given function is f (x) = |x| -|x +1

The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at all points x, such that x < 0

Case II:

If c> 0 then g(c) = c

and \(\lim\limits_{x \to c} g(x) = \lim\limits_{x \to c} x = c\)

\(\lim\limits_{x \to c} g(x) = g(c)\)

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

Case III:

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points

Therefore, h is continuous at all points x, such that x < −1

Case II:

Therefore, h is continuous at x = −1

From the above three observations, it can be concluded that h is continuous at all points of the real line. g and h are continuous functions. Therefore, f = g − h is also a continuous function. Therefore, f has no point of discontinuity.