Derivatives of functions of two variables

Question

Suppose we have a function $f:\mathbb{R}^2\rightarrow \mathbb{R} $ given and say that $$ \frac{df(x,y)}{dx}=0 $$ then of course this means that f is constant with respect to x but does that mean that f is just a constant function?

Answer 1

No. Consider $f(x,y) = y^2+\pi$. Clearly, $$ \frac{\partial f}{\partial x} = \frac{\partial y^2}{\partial x} + 0 = 0 $$ since $y$ is also constant with respect to $x$.

Answer 2

No. Take $$ f(x,y)=y $$ This tells you nothing about how the function changes along the $y$ axis.