I'll start with a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ whose partial derivatives exists at a point, but is not continuous at that point.
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $f(x)=\begin{cases}1, &\mbox{if}& x=0 &\mbox{or if}& y=0\\ 0, &\mbox{otherwise}& \end{cases}$
because $f$ is constant on the $x$ and $y$ axes where it equals $1$, $\frac{\partial f}{\partial x}(0,0)=0$ $\frac{\partial f}{\partial y}(0,0)=0$ But $f$ is not continuous at $(0,0)$ because $\displaystyle \lim_{(x,y)\rightarrow (0,0)}f(x,y)$ does not exist.
Now my issue:
if $f:\mathbb{R} \rightarrow \mathbb{R}$ it is defined by $f(x)=\begin{cases}1, \mbox{if}& x=0 \\ 0, &\mbox{otherwise} \end{cases}$
this function is not continuous but my question is: is there a derivative for this function?- I say no. For the function $f :\mathbb{R}^2\rightarrow \mathbb{R}$ it is possible because $x=y=0$ not in same time? Am I wrong?
Thanks:)