Suppose $f(x_1,x_2,\dots,x_n)$ is a multivariable function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that for all partial derivatives, $1\le i \le n$,
$\frac{\partial f}{\partial x_i}(q_1,q_2,\ldots,q_n) \ge 0$
for all $q_i \ge r_i$.
Is f a non-decreasing function in the set of all points $q_i \ge r_i$ for all $1\le i \le n$?
Edit: By "non-decreasing", I mean that $f(q_1,q_2,\ldots,q_n) \ge f(r_1,r_2,\ldots,r_n) $ if $q_i \ge r_i$ for all i.
Also, is the converse true? Does a function that is 'non-decreasing', by my definition, also have all first partial derivatives positive?