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

#### Solution

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 *c*and 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.