Thetwo functions, g and h, are defined as
Then, f = g − h
Thecontinuity of g and h is examined first.
Clearly, g isdefined for all real numbers.
Let c bea real number.
Case I:
Therefore, g iscontinuous at all points x, such that x < 0
Case II:
Therefore, g iscontinuous at all points x, such that x > 0
Case III:
Therefore, g iscontinuous at x = 0
Fromthe above three observations, it can be concluded that g iscontinuous at all points.
Clearly, h isdefined for every real number.
Let c bea real number.
Case I:
Therefore, h iscontinuous at all points x, such that x < −1
Case II:
Therefore, h iscontinuous at all points x, such that x > −1
Case III:
Therefore, h iscontinuous at x = −1
Fromthe above three observations, it can be concluded that h iscontinuous at all points of the real line.
g and h arecontinuous functions. Therefore, f = g − h isalso a continuous function.
Therefore, f hasno point of discontinuity.