A multivariate function is convex iff it is convex in all axes?

Question

Does the following statement is true?
And if so, how can one prove it?
Given the function $f:R^n->R$
And it is given that for every $x_i\in \bar{x}$ setting $x_j$ $j\neq i$ to zero
The function $f(0,0,0,...,x_i,0,0,...0)$ is convex
The function $f$ is convex

Answer 1

If $g$ is any non-convex function on $\mathbb R$ with $g(0)=0$ then $f(x)=g(x_1)g(x_2)...g(x_n)$ gives a counterexample.