Continuity and Uniform Continuity

Abstract

1 4. The only difference between the two definitions is the order of the quan-tifiers. When you prove f is continuous your proof will have the form Choose x 0 ∈ S. Choose ε > 0. Let δ = δ(x 0 , ε). Choose x ∈ S. Assume |x − x 0 | < δ. · · · Therefore |f (x) − f (x 0)| < ε. The expression for δ(x 0 , ε) can involve both x 0 and ε but must be independent of x. The order of the quanifiers in the definition signals this; in the proof x has not yet been chosen at the point where δ is defined so the definition of δ must not involve x. (The · · · represent the proof that |f (x) − f (x 0)| < ε follows from the earlier steps in the proof.) When you prove f is uniformly continuous your proof will have the form Choose ε > 0. Let δ = δ(ε). Choose x 0 ∈ S. Choose x ∈ S. Assume |x − x 0 | < δ. · · · Therefore |f (x) − f (x 0)| < ε. so the expression for δ can only involve ε and must not involve either x or x 0 . It is obvious that a uniformly continuous function is continuous: if we can find a δ which works for all x 0 , we can find one (the same one) which works for any particular x 0 . We will see below that there are continuous functions which are not uniformly continuous. Example 5. Let S = R and f (x) = 3x + 7. Then f is uniformly continuous on S.