Question

Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x

Answer 1

It is known that if g and h are two continuous functions, then

\[g + h, g - h \text{ and }  g \times h\] 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 → c, then h → 0

`g(c)=sin  c`

`lim_(x->c)g(x)=lim_(x->c) sin  x`

                  `=lim_(h->0) sin (c+h)`

                 `=lim_(h->0)[sin c cos h + cos  c sin h]`

                `=lim_(h->0)(sin  c cos h )+lim_(h->0)(cos c sin h)`

                `=sin  c cos 0+ cos c sin 0`

                `= sin  c+0 `

               `=sin c `

`∴lim_(x->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 → c, then h → 0

h (c) = cos c

`lim_(x->c)h(x)=lim cos  x`

                  `=lim_(h->0) cos  (c+h)`

                  `=lim_(h->0)[cos c cos h - sin  c sin h]`

                 `=lim_(h->0) cos  c cos  h - lim_(h->0)sin  c sin  h`

                `=cos  c cos 0+ cos c sin 0`

                `= cos  c xx1-sin c sin 0`

                `=cos c `

`∴lim_(x->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.