MathDB

We consider a function

f:\mathbb{R} \rightarrow \mathbb{R}

for which there exist a differentiable function

g : \mathbb{R} \rightarrow \mathbb{R}

and exist a sequence

(a_n)_{n \geq 1}

of real positive numbers, convergent to

0,

such that

g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}.

a) Give an example of such a function f that is not differentiable at any point

x \in \mathbb{R}.

b) Show that if

f

is continuous on

\mathbb{R}

, then

f

is differentiable on

\mathbb{R}.

Problem Statement