Question

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

Answer 1

If $f$ is a real function on a subject of the real number and C be a point in the domain of $f$ then $f$ is continuous at C if

$\Rightarrow {lim}_{x\to C}f\left(x\right)=f\left(C\right)$

Step 1 -

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

Let $f\left(x\right)=\cos x$

$\Rightarrow h\left(x\right)=x^2$

$\Rightarrow f\left(h\right).x=f\left(h\left(x\right)\right)=\cos x^2$

Step 2 -

Now $f$ and $h$ both are continues for all $xϵR.$

$\Rightarrow f\left(x\right)=\left(\cos h\right)=\cos x^2$ is also contribute at all $xϵx.$

Hence, solved.