# Differentiable function of two variables

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.

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## 1 Answer

The statement $f \in C^1$ means that $f$ is continuously differentiable.

There is a theorem that says that if each of the partials exists and is continuous at $x$, then the function is continuously differentiable at $x$.

In your case, the partials exist and are continuous on $D$, hence $f \in C^1$.

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