Question 61
Let $f: \mathbb{R} \to \mathbb{R}$ be a function with the property that there are a differentiable function $g: \mathbb{R} \to \mathbb{R}$ and a sequence $(a_n)_{n \ge 1}$ with strictly positive terms and $\lim_{n \to \infty} a_n = 0$, such that
$$
g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n},
$$
for all $x \in \mathbb{R}$.
a) Give an example of such a function $f$ that is not differentiable at any point $x \in \mathbb{R}$.
b) Assume that $f$ is continuous on $\mathbb{R}$. Show that $f$ is differentiable on $\mathbb{R}$.
$$
g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n},
$$
for all $x \in \mathbb{R}$.
a) Give an example of such a function $f$ that is not differentiable at any point $x \in \mathbb{R}$.
b) Assume that $f$ is continuous on $\mathbb{R}$. Show that $f$ is differentiable on $\mathbb{R}$.