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:
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.
