Let $f:\mathbb R^2\to \mathbb R$
$z=f(x,y)$
$f$ is continuous and strictly increasing in both $x,y$.
It is known that continuous in each linear directions does not imply continuity.
But the examples that I found all involve non-monotonic functions.
$$f(x,y)=\begin{cases}\frac{xy}{x^2+y^2},&(x,y)\neq(0,0)\\ 0,&(x,y)=(0,0)\\ \end{cases}$$
I wonder if continuous and increasing in every variable implies continuous?
I think the answer is no
It does.
For example, consider function on $[0, 1] \times [0, 1]$ that is continuous increasing in both $x$ and $y$, and $f(0, 0) = 0$. We want to show that $\forall \epsilon > 0 \exists \delta > 0: x < \delta \wedge y < \delta \rightarrow f(x, y) < \epsilon$.
Because $f$ is continuous in $x$ and $f(0, 0) = 0$, there is $x_0$ s.t. $f(x_0, 0) < \epsilon / 2$. Because $f$ is continuous in $y$, there is $y_0$ s.t. $f(x_0, y_0) < \epsilon$. Now, because $f$ is increasing in $x$ and $y$, if $x < x_0$ and $y < y_0$, then $f(x, y) < \epsilon$. So, taking $\delta = \max(x_0, y_0)$ we get necessary result.