Question

Discuss the continuity of the following functions.
(a) f(x)=sinx+cosx
(b) f(x)=sinx−cosx
(c) f(x)=sinx$\times$cosx

Answer 1

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)=sinx and h(x)=cos x are continuous functions.
Let g(x)=sinx It is evident that g(x)=sinx 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)=sinc
$\lim_{x\rightarrow c}g(x)$=$\lim_{x\rightarrow c}sin\,x$
=$\lim_{h\rightarrow 0}sin(c+h)$
=$\lim_{h\rightarrow 0}$[sin c cos h+cos c sin h]
=sin c cos 0+cos c sin 0
=sin c+0=sin c
∴$\lim_{x\rightarrow c}g(x)$=$g(c)$
Therefore, g is a continuous function.
Let h(x)=cos x It is evident that h(x)=cosx 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)=cosc
$\lim_{x\rightarrow c}$h(x)=$\lim_{x\rightarrow c}$ cos x
=$\lim_{h\rightarrow 0}$cos(c+h)
=$\lim_{h\rightarrow 0}$[cosccosh-sincsinh]
=$\lim_{h\rightarrow 0}$cos c cos h-$\lim_{h\rightarrow 0}$ sin c sin h
=cos c cos 0-sin csin 0
=cos c$\times$1-sin c$\times$0
=cos c
∴$\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)=sinx+cosx is a continuous function
(b)f(x)=g(x)-h(x)=sinx−cosx is a continuous function
(c)f(x)=g(x)×h(x)=sinx×cosx is a continuous function