The Intermediate Value Theorem. This contradictions the definition of continuity of f . Let x n be any sequence that converges to x . Theorem 3.14 tells us that functions that are continuous on a closed interval will also be uniformly continuous on that interval Proof of Theorem 3.14 Suppose that f is not uniformly continuous. As we have seen in other examples, given any real number x0, since the rational numbers are dense we can always define a sequence of rational numbers that converges to x 0 .
Every real number is a limit point of the set of irrational numbers, since they are dense also.
Example 3.12 and 3.13 give examples of functions whose limits exist at x 0 but are not continuous at x 0 . The Sum, Product, and Quotient Rules for limits . Skipping Theorem 3.16 for now.