Question

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

Answer 1

Let $f\left(x\right)=\cos x^2$ and $a$ be any real number.

Now,

LHL at $x=a$

$\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a-h)=\underset{h\to 0}{lim}\cos {(a-h)}^2=\cos a^2$

RHL at $x=a$

$\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a+h)=\underset{h\to 0}{lim}\cos {(a+h)}^2=\cos a^2$

Also,

$f\left(a\right)=\cos a^2$

$\Rightarrow \underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$

Hence $f\left(x\right)$ is continuous at $x=a$.

Since $a$ is an arbotrary real number, thus $f\left(x\right)$ is continuous at everywhere.