Show that the function defined by f (x) = | cos x| is a continuous function.

Question

Philoid · Accepted Answer

It is given that f(x) = |cosx|

The given function f is defined for real number and f can be written as the composition of two functions, as

f = goh, where g(x) = |x| and h(x) = cosx

First we have to prove that g(x) = |x| and h(x) = cosx are continuous functions.

g(x) = |x| can be written as

Now, g is defined for all real number.

Let k be a real number.

Case I: If k < 0,

Then g(k) = -k

And

Thus,

Therefore, g is continuous at all points x, i.e., x > 0

Case II: If k > 0,

Then g(k) = k and

Thus,

Therefore, g is continuous at all points x, i.e., x < 0.

Case III: If k = 0,

Then, g(k) = g(0) = 0

Therefore, g is continuous at x = 0

From the above 3 cases, we get that g is continuous at all points.

h(x) = cosx

We know that h is defined for every real number.

Let k be a real number.

Now, put x = k + h

If

= coskcos0 – sinksin0

= cosk × 1 – sin × 0

= cosk

Thus, h(x) = cosx is continuous function.

We know that for real valued functions g and h, such that (goh) is defined at k, if g is continuous at k and if f is continuous at g(k),

Then (fog) is continuous at k.

Therefore, f(x) = (gof)(x) = g(h(x)) = g(cosx) = |cosx| is a continuous function.