If function is concave in each of its variables keeping the rest fixed, is the function concave.

Question

Let $f(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$

If the function is concave in each of its variables keeping the other fixed, is the function $f(x,y)$ concave?

That is, if we have:

$f(x,c)$ $\quad \forall c \in \mathbb{R}$ is concave w.r.t $x$

$f(c,y)$ $\quad \forall c \in \mathbb{R}$ is concave w.r.t $y$

Is $f(x,y)$ concave?

Answer 1

No. For instance $f(x,y)=xy$.