I have to determine if $\displaystyle f(x,y) = \frac{x}{y} + \frac{y}{x}$ is differentiable at all points in its domain and of class $\displaystyle C^1$

$\displaystyle \frac{\partial f}{\partial x} = \frac{1}{y} - \frac{y}{x^2}$

$\displaystyle \frac{\partial f}{\partial y} = \frac{1}{x} - \frac{x}{y^2}$

seems that if $\displaystyle (x,y)=(0,0)$ then its not continuous or differentiable

and if $\displaystyle x=0$ or $\displaystyle y=0$ then one or both partials fail to be continuous or differentiable

so no, its not of class $\displaystyle C^1$

correct?