Is the multi variable function convex when it is convex for all of it's arguments?

Question

Let $f=f(x,y)\in C^1(\Omega)$ for some convex domain $\Omega\in\mathbb{R}^2$. Suppose $x\mapsto f(x,y)$ is convex $\forall y$ possible and $y\mapsto f(x,y)$ is also convex $\forall x$ possible. Is $f$ convex on $\Omega$?

I think it is not true, but cannot come up with proper counterexample.

Answer 1

Example: $f(x,y) = xy$.

For each fixed $y$ it is linear in $x$ and therefore convex in $x$. For each fixed $x$, similarly.

But along the line $x+y=0$ it is not convex.

$f(1,-1) = -1$, $f(-1,1) = -1$, but the midpoint $f(0,0) = 0$ is not below $-1$.