Find the domain of a function $f(x,y)=\log_xy$. Does $f\in C^1$? (I don't know if this symbol is common - it means that $f$ is at least once differentiable, and its first derivative is continuous).
from basic facts the domain is: $D=\mathbb{R}^2 \setminus \left\{ (x,y) : x=1 \vee y\le 0 \vee x\le 0 \right\}$
partial derivatives (maybe it will useful):
$\frac{\partial}{\partial x}f(x,y)=-\frac{\ln y}{x\ln^2 x}$
$\frac{\partial}{\partial y}f(x,y)=\frac{1}{y\ln x}$
but I don't understand what is necessary for $f\in C^1$. Do I just need to check if partial derivatives are both continuous for all $x,y\in D$?
I would be very grateful for help.