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) = 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
Therefore, 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
Therefore, h is a continuous function.
Therefore, it can be concluded that
(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function
(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function
(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function
