Question

Show that the function defined by f (x) = cos (x²) is a continuous function.

Answer 1

The given function is f (x) = cos (x²)
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) = cos x and h (x) = x²

Therefore, g (x) = cos x is continuous function. 

h (x) = x² 

Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k²

Therefore, h 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.