Question

Show that the function defined by f(x) = cos $x^2$ is a continuous function.

Answer 1

Let g(x) =$x^2$ and h(x) = cos x

Now, h(x) is a polynomial function, so it is continuous for all $xϵR$

g(x) is a cosine function, so it is continuous function in its domain i.e., $xϵR$

$\therefore$ (goh)(x)=g[h(x)]=$g\left(x^2\right)$=cos $x^2$

Since, g(x) and h(x) are both continuous functions for all $xϵR$ so, composition of g(x) and h(x) is also a continuous function for all $xϵR$

Thus, f(x) =con$\left(x^2\right)$ is a continuous function for all