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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
Solution
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.
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