Show that the function defined by f (x) = cos (x2) is a continuous function.
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
`[∴(goh)(x) = g(h(x)) = g(x^2) = cos(x^2) = f(x)]`
It has to be first proved that g (x) = cos x and h (x) = x2 are continuous functions.
It is evident that g is defined for every real number.
Let c be a real number.
Then, g (c) = cos c
Put x = c + h
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.
Therefore, `f(x) = (goh)(x) = cos(x^2)` is a continuous function.