Show that the function defined by f(x) = |cos x| is a continuous function.
The given function is f(x) = |cos x|
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g o h, where g(x) = |x| and h(x) = cos x
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 cand if f is continuous at g (c), then (f o g) is continuous at c.
Therefore, f(x) =(goh)(x) = g(h(x)) = g(cos x) = |cos x| is a continuous function.