let $f : \mathbb{R}^3 \to \mathbb{R}$ be a $C^1$ function. and we have $\frac{\partial f}{\partial x }= \frac{\partial f}{\partial y} = 0$. Prove that function $g(x,y) = f(x,y,a)$ for fixed real number $a$ is a constant function.
I really don't have any idea to solve this. Although I know it must be straight forward. Is this enough to say that derivative of $g$ is zero? If yes, Should I use partial derivative?
Since $f$ is a $C^1$ function, then so is $g$. And since $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$ everywhere, then $\frac{\partial g}{\partial x}=\frac{\partial g}{\partial y}=0$ everywhere. But then the derivative of $g$ is $0$ everywhere, and therefore $g$ is constant.