Math 312 Sections 1 2 Lecture Notes Section 13 5 Uniform continuity Definition Let S be a non empty subset of R We say that a function f S R is uniformly continuous on S if for each 0 there is a real number 0 such that f x f y for all x y S with x y where depends only on Remarks a Uniform continuity is a property involving a function f and a set S on which it is defined It makes no sense to speak of f being uniformly continuous at a point of S except when S consists of a single point b Remember that the in the usual definition of ordinary continuity depends both on and the point x0 at which continuity is being tested So uniform continuity implies continuity But the converse is false as we will see below in Example 5 Example 1 Let f x 5x 8 Prove that f is uniformly continuous on R Solution We have f x f y 5 x y Given 0 we choose 5 then f x f y for all x y R with x y Therefore f is uniformly continuous on R Example 2 Let f x x2 on I 0 1 Prove that f is uniformly continuous on I Solution We have f x f y x2 y 2 x y x y 2 x y for x y in 0 1 Given 0 we choose 2 then f x f y for all x y 0 1 with x y Therefore f is uniformly continuous on I 0 1 Example 3 Let f x 1 x2 defined on x in I 3 Show that f is uniformly continuous on I 3 Solution We have x2 y 2 2 x y x y 1 1 f x f y x y x y x2 y 2 x2 y 2 xy 2 x2 y 27 Given 0 we choose 27 then f x f y for all x y 3 2 with x y Therefore f is uniformly continuous on 3 Example 4 Let f x cos x Prove that f is uniformly continuous on R Solution We have seen that cos x cos y x y for all x y R Given 0 we choose then f x f y for all x y I with x y Therefore f is uniformly continuous on R Example 5 Let 1 defined on I 0 1 x Show that f is not uniformly continuous on I Solution The function f x x1 is certainly continuous on the open interval 0 1 But it is not uniformly continuous on 0 1 Indeed let x and y be elements of 0 1 then f x f x f y y x 1 1 x y xy Here the key idea is the fact that x y xy might approach infinity if x and y approach zero and therefore f x f y will be bigger than any chosen Now we will formalize this idea We pick 1 Without loss of generality we can suppose that 0 1 Set x 2 and y then f x f y 2 1 1 Therefore the function is not uniformly continuous on 0 1 Theorem Let I be a compact interval Then f I R is continuous on I if and only if f is uniformly continuous