Question

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

Answer 1

Let's assume that $h\left(x\right)=x^2$ and $g\left(x\right)=\cos x$

So $f\left(x\right)=\left(goh\right)=\cos \left(x^2\right)$

Let's prove that $g\left(x\right)=\cos \left(x\right)$ is continuous in it's domain.

Let $c$ be a real number, put $x=c+h$

So if $x\Rightarrow c$ then it means that $h\Rightarrow 0$

$f\left(c\right)=\cos \left(c\right)$

$\underset{x\Rightarrow c}{lim}$ $f\left(x\right)$ = $\underset{x\Rightarrow c}{lim}$ $\cos \left(x\right)$

Put $x=h+c$

And as mentioned above, when $x\Rightarrow c$ then it means that $h\Rightarrow 0$

Which gives us $\underset{h\Rightarrow 0}{lim}$ $\cos (c+h)$

Expanding $\cos (h+c)$ = $\cos \left(h\right)\cos \left(c\right)-\sin \left(h\right)\sin \left(c\right)$

Which gives us $\underset{h\Rightarrow 0}{lim}$ $\cos \left(h\right)\cos \left(c\right)-\sin \left(h\right)\sin \left(c\right)$

$=\cos \left(c\right)\cos \left(0\right)-\sin \left(c\right)\sin \left(0\right)$

$=\cos \left(c\right)$

And this proves that $\cos \left(x\right)$ is continuous all across its domain

Now let's see if $x^2$ is continues in its domain

$\underset{x\Rightarrow h}{lim}$ $h\left(x\right)$ = $\underset{x\Rightarrow h}{lim}$ $x^2=h^2=h\left(c\right)$ and this proves that $h\left(x\right)$ is continuos in its domain.

So by theorem: If function $f$ and function $g$ are continuous then $fog$ is also continuous.

There for $\cos \left(x^2\right)$ is continuous across its domain.