4 is continuous at x g Proof We are investigating continuity of f g at x c Clearly it is defined at x c We have 114 lim f g x MATHEMATICS provided g c 0 lim f x g x X C lim f x lim g x X C X C f c g c f g c Hence f g is continuous at x c Proofs for the remaining parts are similar and left as an exercise to the reader Rationalised 2023 24 Remarks i As a special case of 3 above iffis a constant function i e f x for some real number 2 then the function 2 g defined by 2 g x 2 g x is also continuous In particular if 1 the continuity of fimplies continuity of f ii As a special case of 4 above if f is the constant function f x 2 then the 2 2 2 function defined by x is also continuous wherever g x 0 In g g x g particular the continuity of g implies continuity of g by definition of f g by the theorem on limits as fand g are continuous by definition of f g The above theorem can be exploited to generate many continuous functions They also aid in deciding if certain functions are continuous or not The following illustrate this examples Example 16 Prove that every rational function is continuous Solution Recall that every rational function fis given by q x 0 p x q x f x where p and q are polynomial functions The domain offis all real numbers except points at which qis zero Since polynomial functions are continuous Example 14 fis continuous by 4 of Theorem 1 Example 17 Discuss the continuity of sine function Solution to see this we use the following facts We have not proved it but is intuitively clear from the graph of sin x near 0 Now observe that f x sin x is defined for every real number Let c be a real xc we know that h 0 Therefore number Put x c h lim f x lim sin x X C X C oblishe lim sin c h h 0 lim sinc cos h cos c sin h h 0 lim Isin c cos hl lim cosc sin hl