This may be too easy, but:

Is there a function $f$ on the first quadrant of $\mathbb R^2$ such that $$ f(x,1)=x,\qquad f(x,0)=0, $$ and $f$ is convex or concave?

Note there is no solution of the form $f(x,y)=x\cdot g(y)$, since (i) $-f$ is convex iff $f$ is concave, and (ii) the Hessian is $$ H=\left( \begin{matrix} 0& g'(y) \\ g'(y)&x\cdot g''(y) \end{matrix} \right) $$ which is positive semi-definite only if $$ -(g'(y))^2\ge 0 $$ which would mean $g(y)$ is constant.