How to prove that a set is not path-connected?

Question

Mathematics

Let's say we have a plane R^2. Point coordinates (X, Y).
The range of the function f(x, y) = 1/(xy) will be R^2\{(x, y): (xy) = 0}, that is, the entire plane except for the straight line x=y. It is clear that such a set will not be linearly connected, since it is impossible to set a continuous mapping between points above this line and below it. But how can one prove it rigorously? Without any "obvious" and so on and so forth.
Picture:

Answer 1

Used the intermediate value theorem for continuous mappings

Answer 2

This space is not connected, therefore it is not path-connected.