let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) =\begin{cases} \dfrac{\arctan(x)}{x} & \text{ if } x=0 \\
\\
f(0)=1 & \end{cases} $$
$f$ is not differentiable at $0$
$f(x)\mathrel{\underset{_0}{=}}1+o(x)$ and $f$ is differentiable at $0$ and $f'(0)=0$
${\displaysty...
let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) =\begin{cases} \dfrac{\arctan(x)}{x} & \text{ if } x=0 \\
\\
f(0)=1 & \end{cases} $$
$f$ is not differentiable at $0$
$f(x)\mathrel{\underset{_0}{=}}1+o(x)$ and $f$ is differentiable at $0$ and $f'(0)=0$
${\displaysty...
