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 $g . h$ 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 $h \rightarrow 0$

$g(c)=\sin c$

$\begin{aligned} \lim _{x \rightarrow c} g(x) &=\lim _{x \rightarrow c} \sin x \\ &=\lim _{h \rightarrow 0} \sin (c+h) \\ &=\lim _{h \rightarrow 0}[\sin c \cos h+\cos c \sin h] \\ &=\lim _{h \rightarrow 0}(\sin c \cos h)+\lim _{h \rightarrow 0}(\cos c \sin h) \\ &=\sin c \cos 0+\cos c \sin 0 \\ &=\sin c+0 \\ &=\sin c \end{aligned}$

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

Therefore, 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

$\begin{aligned} \lim _{x \rightarrow c} h(x) &=\lim _{x \rightarrow c} \cos x \\ &=\lim _{h \rightarrow 0} \cos (c+h) \\ &=\lim _{h \rightarrow 0}[\cos c \cos h-\sin c \sin h] \\ &=\lim _{h \rightarrow 0} \cos c \cos h-\lim _{h \rightarrow 0} \sin c \sin h \\ &=\cos c \cos 0-\sin c \sin 0 \\ &=\cos c \times 1-\sin c \times 0 \\ &=\cos c \end{aligned}$

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

Therefore, h is a continuous function.

Therefore, it can be concluded that

(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function

(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function

(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function