f ( c ) and hence f is a continuous function. Remark A similar proof may be given for the continuity of cosine function. Example Prove that the function defined by f ( x ) = tan x is a continuous function. Solution The function f ( x ) = tan x = sin x .
This is defined for all real numbers such that cos x ≠ , i.e., x ≠ ( n + ) π . 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. An interesting fact is the behaviour of continuous functions with respect to composition of functions.
Recall that if f and g are two real functions, then ( f o g ) ( x ) = f ( g ( x )) is defined whenever the range of g is a subset of domain of f . The following theorem (stated without proof) captures the continuity of composite functions. Theorem Suppose f and g are real valued functions such that ( f o g ) is defined at c . If g is continuous at c and if f is continuous at g ( c ), then ( f o g ) is continuous at c .
The following examples illustrate this theorem. Example 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 , it can be deduced that f is a continuous function. Example Show that the function f defined by f ( x ) = | – x + | x ||, where x is any real number,