Let $f : B_1(0) → R$ be a function with the charactheristic that the partial derivatives of $f$ exists in all points in $B_1(0)$. Show that $f$ is constant if and only if $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = 0$ everywhere in $B_1(0)$.
My first idea was to show this both ways. I would argue that the derivatives always being 0 entails that it is constant. If the derivatives are always the same, then $f$ would also have to always be the same
I am, however, more confused about how i would show the other end of the proof.
