3. Prove that a composition of uniformly continuous functions is uniformly conti

Question

3. Prove that a composition of uniformly continuous functions is uniformly continuous.

Explanation / Answer

uniformly continuous: f(x) = sqrt(x) on the closed interval [0, 1] (and therefore also the open interval (0, 1)) f(x) = x on all real numbers f(x) = any constant function f(x) = x^2 on the closed interval [0, 1] not uniformly continuous: f(x) = 1/x on the open interval (0, 1) f(x) = 1/x^n on the open interval (0, 1), where n >= 1 There are some wonderful theorems concerning uniform continuity: If f is a continuous function of a compact set, it is uniformly continuous on that set. If f is differentiable on an interval (a, b), and if the derivative f' is bounded on that interval (a, b), then f is uniformly continuous. Note that the converse of the latter theorem does not hold in general, since f(x) = sqrt(x) is uniformly continuous on (0, 1), but its derivative is unbounded as x approaches 0. Hope this helps.

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.