We known that g and k are two continuous functions, then,

g + k, g – k and g.k are also continuous.

First we have to prove that g(x) = sinx and k(x) = cosx are continuous functions.

Now, let g(x) = sinx

We know that g(x) = sinx is defined for every real number.

Let h be a real number. Now, put x = h + k

So, if

g(h) = sinh

= sinhcos0 + coshsin0

= sinh + 0

= sinh

Thus,

Therefore, g is a continuous function…………(1)

Now, let k(x) = cosx

We know that k(x) = cosx is defined for every real number.

Let h be a real number. Now, put x = h + k

So, if

Now k(h) = cosh

= coshcos0 - sinhsin0

= cosh - 0

= cosh

Thus,

Therefore, k is a continuous function……………….(2)

So, from (1) and (2), we get,

(a) f(x) = g(x) + k(x) = sinx + cosx is a continuous function.

(b) f(x) = g(x) - k(x) = sinx - cosx is a continuous function.

(c) f(x) = g(x) × k(x) = sinx × cosx is a continuous function.