Question

true/false? prove..
Every function differentiable on [a, b] is bounded on [a, b]

Answer 1

Answer on Question #69118 – Math – Real Analysis

Question

True/false? Prove.

Every function differentiable on $[a, b]$ is bounded on $[a, b]$.

Solution

Let us start from the fact that if function $f$ is differentiable at $x_0$ then $f$ is continuous at $x_0$ (see https://proofwiki.org/wiki/Differentiable_Function_is_Continuous).

Since function $f$ is differentiable on $[a, b]$ (i.e. $f$ is differentiable at any $x_0 \in [a, b]$) then $f$ is continuous at any $x_0 \in [a, b]$, i.e. $f$ is continuous on $[a, b]$.

Since $f$ is continuous on $[a, b]$ then $f$ is bounded on $[a, b]$ (see http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L21.html).

Answer: True.

Answer provided by https://www.AsignmentExpert.com