This function f is defined for every real number and f can be written as the composition of two functions 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. h (x) = cos x It is evident that h (x) = cos x is defined for every real number. Let c be a real number. Put x = c + h If x → c, then h → 0 h (c) = cos c
Therefore, h (x) = cos x is a continuous function. It is known that for real valued functions g and h,such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
