Monotonicity of function of two variables

There is no general definiton (as mentioned in the comments, there is no total order on $\mathbb{R}^2$, which would be required for a canonical definition of monotonicity of bivariate functions).

Two possible definitions, though: let $$f\colon (x,y)\in\mathbb{R}^2\mapsto f(x,y)$$ be your function.

• First definition: $f$ is said to be monotone (non-decreasing) if for all fixed $x_0, y_0 \in \mathbb{R}$, the two functions $f_{x_0}\colon y\in\mathbb{R}\mapsto f(x_0,y)$ and $f_{y_0}\colon x\in\mathbb{R}\mapsto f(x,y_0)$ are monotone (non-decreasing). (i.e., monotonicity wrt both projections)

• Second definition: $f$ is said to be monotone (non-decreasing) if for all fixed $(x, y),(x', y') \in \mathbb{R}^2$, $$(x \leq x' \text{ and } y \leq y' ) \Rightarrow f(x,y) \leq f(x',y')$$ (i.e., monotonicity wrt a partial order on $\mathbb{R}^2$)

Edit: I had originally written the two definitions are not equivalent. They are, as Ij Huij's answer below shows. I can't reconstruct what I had in mind at the time, but to be charitable it's probably either very contrived or wrong...

I wanted to add this as a comment for the first answer, but I cannot put comments. The first and the second definition in this answer are equivalent.

If $f$ satisfies the first definition then it satisfies the second. We can see that by sandwiching $f(x',y)$ (or $f(x,y')$) between $f(x,y)$ and $f(x',y')$.

If $f$ satisfies the second definition, by once setting $x = x'$ and once setting $y = y'$ we can see that it is monotone with respect to both projections.

In general, let $\preceq$ be the elementwise inequality on $\mathbf{R}^n$ ($x \preceq y$ if and only if $x_i \leq y_i$ for all $i$). Now, a function $f: \mathbf{R}^n \mapsto \mathbf{R}$ is monotone with respect to this partial ordering when:

$$x \preceq y \Rightarrow f(x) \leq f(y).$$ This is equivalent to requiring that for all $x \in \mathbf{R}^n$ $$f(x) \leq f(x + \gamma e_i) \quad \text{for all } \gamma \geq 0 \text{ and }i \in\{1,2,...,n\}, $$ where $\{e_1,\ldots,e_n\}$ is the standard basis for $\mathbf{R}^n$.