Several times I faced the following

**Definition 1:** A function $f: \mathbb{R^n}\mapsto \mathbb{R}$ is called $k$ times differentiable at $x\_0$ iff all the partial derivatives of $f$ of order $k-1$ are differentiable at $x\_0$.

Besides this definition, more popular is the other 

**Definition 2:** A function $f: \mathbb{R^n}\mapsto \mathbb{R}$ is called $k$ times differentiable at $x\_0$ iff *$f$ is $k-1$ times differentiable in a neighborhood of $x\_0$* and all the partial derivatives of $f$ of order $k-1$ are differentiable at $x\_0$.

It's obvious, that in case $n=1$ these two definitions coincide. Besides that, it can be shown that Def. 2 coincides with general Frechet $k$-times differentiability and obviously it's not weaker than Def 1. On the other hand, it can be shown that Def. 1 is sufficient to prove general Young's theorem. 
In this regard, the following question arose:

1. Does anyone know an example of a function (say 2 times differentiable of 2 variables) which satisfies the Def. 1, but not the Def. 2? I have some ideas of constructing such an example with the use of Sobolev mollifications, but it's quite complicated and ugly ….
2. If such an example exists, is Taylor theorem with Peano remainder valid for the functions which satisfy the Def. 1?