Frechet Differentiability versus Strict Differentiability We recall two concepts of differentiability of a mapping. Let $X, Y$ be real Banach spaces and $f: X\rightarrow Y$
be a mapping. 


*

*$f$ is said to be Frechet differentiable at $\bar{x}$ if there
exists a linear continuous operator $\nabla f(\bar{x}): X\rightarrow Y$ such that
$$
\lim_{x\rightarrow \bar{x}}\frac{f(x)-f(\bar{x})-\nabla f(\bar{x})(x-\bar{x})}{\|x-\bar{x}\|}=0.
$$

*$f$ is said to be strictly differentiable at $\bar{x}$ if $f$ is Frechet differentiable
at $\bar{x}$ and 
$$
\lim_{\substack{x\rightarrow \bar{x}\\ u\rightarrow \bar{x}}}\frac{f(x)-f(u)-\nabla f(\bar{x})(x-u)}{\|x-u\|}=0.
$$
It is known that if $f$ is continuously Frechet differentiable in a neighborhooh of $\bar{x}$
 then $f$ is strictly differentiable at this point but not vice versa.
I would like to find a mapping $f$ such that $f$ is strictly differentiable at $\bar{x}$
but it is not differentiable at points near $\bar{x}$.

$f: [-1,1] \rightarrow \mathbb R , \bar x=0$, defined by
$f(x):=\begin{cases} x^2 & x=1/k , k\in \mathbb N \\ 0 & x=0\\ \text{linear} & \text{otherwise} \end{cases}$
