For a function of one variable just looking at the graph I can see that the condition for a one-to-one analytic function is:
$$y=f(x)$$
is one-to-one if we have the condition:
$f'(x)>0$ for all $x$. i.e. the function is always increasing.
Then one might write $f'(x) = g(x)^2$ for some function $g$. Then we could write that a general function $f$ is a one-to-one:
$$f(x)\equiv \int^x_0 g(x)^2 dx$$
How would we generalise this to multiple variables? e.g. $y_n = f_n(x_1,..x_n)$