Answer :

It is given function is f(x) = cos (x2)

This function f is defined for every real number and f can be written as the composition of two function as,


f = goh, where, g(x) = cosx and h(x) = x2


First we have to prove that g(x) = cosx and h(x) = x2 are continuous functions.


We know that g is defined for every real number.


Let k be a real number.


Then, g(k) =cos k


Now, put x = k + h


If






= coskcos0 – sinksin0


= cosk × 1 – sin × 0


= cosk



Thus, g(x) = cosx is continuous function.


Now, h(x) = x2


So, h is defined for every real number.


Let c be a real number, then h(c) = c2




Therefore, h is a continuous function.


We know that for real valued functions g and h,


Such that (fog) is continuous at c.


Therefore, f(x) = (goh)(x) = cos(x2) is a continuous function.


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