The paragraph leaves out an important constraint: it’s only for functions that map some compact convex set to itself. So, the theory doesn’t apply to the siblin

The paragraph leaves out an important constraint: it’s only for functions that map some compact convex set to itself. So, the theory doesn’t apply to the sibling commenter’s example of f(x) = x + 1 because it doesn’t map a compact set to itself. One example of a compact convex set is a circle. Imagine some continuous function that maps a circle to a circle. An easy example is rotation. For rotation, the center stays in the same place. Now imagine some more complicated function that might do rotation, stretching, spiral distortions, whatever you can imagine. But it has to be continuous, meaning you can’t go off one edge and come back on the other side (among other things). Try to draw, or even just imagine a transformation like this and you’ll find that there’s always some point that stays in the same place. For a 3D object like a glass it’s hard to see the same thing intuitively, but Brouwer’s theorem says that it holds (and it also holds for higher-dimensional sets).