Lebl-Basic-Analysis-I · Problem 4.3.8

The Derivative

Mathematics · taylor-converse-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

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

, that is,

f

goes to zero at

0

faster than

x^2

, and while

f'(0)

exists,

f''(0)

does not.