Basic Analysis I — Real Analysis · Problem 4.3.8

The Derivative — Jiri Lebl

Mathematics — Real Analysis · converse-taylor-counterexample

Problem

(Challenging): Show that a simple converse to Taylor's theorem does not hold. Find a function

f : \mathbb{R} \to \mathbb{R}

with no second derivative at

x = 0

such that

\bigl|f(x)\bigr| \leq \bigl|x^3\bigr|

, that is,

f

goes to zero at

0

faster than

x^2

, and while

f'(0)

exists,

f''(0)

does not.