If $f(x,y)$ is $C^{\infty}$ in $x$ and $y$ respectively, can we say that $f$ is $C^{\infty}$ in $x$ and $y$ simultaneously?

Question

$f$ is a function from $\mathbb{R}^2\to \mathbb{R}$. I feel like this can't be true but I need to come up with some counterexample, but I'm not sure how. Can you please help?