Question

Examine the continuity of f, where f is defined by

f (x) = sin x - cos x, if x ≠0

f (x) = -1  if x = o

Answer 1

It is evident that f is defined at all points of the real line. Let c be a real number.
Case I:

Therefore, f is continuous at all points x, such that x ≠ 0
Case II:

Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of the real line. Thus, f is a continuous function.