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