Example 18 Prove that the function defined by f x tan x is a continuous function Solution The function f x tan x sin x COS X TU that cos x 0 i e x 2n 1 2 We have just proved that both sine and cosine functions are continuous Thus tan x being a quotient of two continuous functions is continuous wherever it is defined This is defined for all real numbers such An interesting fact is the behaviour of continuous functions with respect to composition of functions Recall that if fand g are two real functions then fog x f g x is defined whenever the range of g is a subset of domain off The following theorem stated without proof captures the continuity of composite functions Theorem 2 Suppose fand g are real valued functions such that fog is defined at c If g is continuous at c and if fis continuous at g c then fog is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f x sin x is a continuous function Solution Observe that the function is defined for every real number The function f may be thought of as a composition g o h of the two functions g and h where g x sin x and h x x Since both g and h are continuous functions by Theorem 2 it can be deduced that fis a continuous function Example 20 Show that the function f defined by f x 11 x x where x is any real number is a continuous function Solution Define 116 MATHEMATICS h 1 x x 1 x x f x In Example 7 we have seen that h is a continuous function Hence g being a sum of a polynomial function and the modulus function is continuous But then f being a composite of two continuous functions is continuous g x 1 x x and h by h x x for all real x Then hog x h g x Rationalised 2023 24 EXERCISE 5 1 1 Prove that the function f x 5x 3 is continuous at x 0 at x 3 and at x 5 2 Examine the continuity of the function f x 2x 1 at x 3 3 Examine the following functions for continuity a f x x 5 b f x 1 x 5 x 5