Question

Discuss the continuity of the following function : $f\left(x\right)=\sin x.\cos x$

Answer 1

Here clearly both $\sin x$ and $\cos x$ are defined in their domain.

Let's assume that $g\left(x\right)=\sin x$ and $f\left(x\right)=\cos x$

=> Let's first prove that $g\left(x\right)$ 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 $h\Rightarrow 0$

$\underset{x\Rightarrow c}{lim}$ $g\left(x\right)$ = $\underset{x\Rightarrow c}{lim}$ $sin\left(x\right)$

Put $x=h+c$

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

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

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

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

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

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

So here we get,

$\underset{x\to c}{lim}$ $g\left(x\right)$ = $\underset{x\Rightarrow c}{lim}$ $\sin \left(x\right)$ = $\sin \left(c\right)=g\left(c\right)$

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

=> Let's prove that $f\left(x\right)=\cos \left(x\right)$ 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 $\underset{h\Rightarrow 0}{lim}$ $\cos \left(h\right)\cos \left(c\right)-\sin \left(h\right)\sin \left(c\right)$

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

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

This gives us

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

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

=> So by theorem, if function $f$ and function $g$ are continous, then $f.g$ is also continous.

There for $\sin \left(x\right).\cos \left(x\right)$ is continous.