Chain rule with multiple variables

My real analysis textbook asks me to

Let $\displaystyle h(u,v)=f(u+v,u-v)$. Show that $\displaystyle f_{xx}-f_{yy}=h_{uv}$.

It looks like a fairly simple problem, at the end of the chapter on the chain rule. However, when I apply the chain rule, I get

$\displaystyle h_{uv}=f_{xx}-f_{yy}+f_{yx}-f_{xy}$

Now, if I can show that either $\displaystyle f_{xy}$ or $\displaystyle f_{yx}$ is continuous on $\displaystyle \mathbb{R}$, then I can let $\displaystyle f_{xy}=f_{yx}$ and the conclusion follows. But how do I prove that with so little information? Or am I going about this entirely the wrong way?

Thanks!