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.)

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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