Convexity in multivariable function

Question

If a function $f(x,y)$ is convex in $x \in \mathbb{R}$, and also convex in $y \in \mathbb{R}$ then is $f$ a convex function in $\mathbb{R}^2$? I think no, for example $f(x,y) = xy$ it is convex in $x$, and $y$ but the function is not convex. Am i correct?