The assignment is worth 30 homework points. (a) [10 points] Sketch the graph of a function which is continuous everywhere in R but has infinitely many points where it is not differentiable. (Hint: consider how the absolute value function fails to be differentiable at 0.) (b) [20 points] Let f : R → R be a function and suppose that |f(x) f(a)|≤ (x − a)² for every x, a Є R. Prove that f is constant. (Note: you do not know a priori that f is continuous, let alone differentiable. However, if you can show that f is differentiable then the fact that ƒ' = 0 on an interval implies that ƒ is constant may be helpful.)

The assignment is worth 30 homework points. (a) [10 points] Sketch the graph of a function which is continuous everywhere in R but has infinitely many points where it is not differentiable. (Hint: consider how the absolute value function fails to be differentiable at 0.) (b) [20 points] Let f : R → R be a function and suppose that |f(x) f(a)|≤ (x − a)² for every x, a Є R. Prove that f is constant. (Note: you do not know a priori that f is continuous, let alone differentiable. However, if you can show that f is differentiable then the fact that ƒ' = 0 on an interval implies that ƒ is constant may be helpful.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section: Chapter Questions

Problem 35RE

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