Discuss the continuity of the following functions:

It is known that if $g$ and $h$ are two continuous functions, then $g+h, g-h$ and $g \times h$ are also continuous.

It has to proved first that $g(x)=\sin x$ and $h(x)=\cos x$ are continuous functions.

Let $g(x)=\sin x$

It is evident that $g(x)=\sin x$ is defined for every real number.

Let $c$ be a real number. Put $x=c+h$

If $x \rightarrow c$, then $h \rightarrow 0$

$g(c)=\sin c$

$\lim _{x \rightarrow c} g(x)=\lim _{x \rightarrow c} \sin x$

$=\lim _{h \rightarrow 0} \sin (c+h)$

$=\lim _{h \rightarrow 0}[\sin c \cos h+\cos c \sin h]$

$=\lim _{h \rightarrow 0}(\sin c \cos h)+\lim _{h \rightarrow 0}(\cos c \sin h)$

$=\sin c \cos 0+\cos c \sin 0$

$=\sin c+0$

$=\sin c$

$\therefore \lim _{x \rightarrow c} g(x)=g(c)$

So, $g$ is a continuous function.

Let $h(x)=\cos x$

It is evident that $h(x)=\cos x$ is defined for every real number.

Let $c$ be a real number. Put $x=c+h$

If $x \rightarrow c$, then $h \rightarrow 0$

$h(c)=\cos c$

$\lim _{x \rightarrow c} h(x)=\lim _{x \rightarrow c} \cos x$

$=\lim _{h \rightarrow 0} \cos (c+h)$

$=\lim _{h \rightarrow 0}[\cos c \cos h-\sin c \sin h]$

$=\lim _{h \rightarrow 0} \cos c \cos h-\lim _{h \rightarrow 0} \sin c \sin h$

$=\cos c \cos 0-\sin c \sin 0$

$=\cos c \times 1-\sin c \times 0$

$=\cos c$

$\therefore \lim _{x \rightarrow c} h(x)=h(c)$

So, $h$ is a continuous function.

Therefore, it can be concluded that

(i) $f(x)=g(x)+h(x)=\sin x+\cos x$ is a continuous function.

(ii) $f(x)=g(x)-h(x)=\sin x-\cos x$ is a continuous function.

(iii) $f(x)=g(x) \times h(x)=\sin x \cos x$ is a continuous function.