Fixed-point theorems are theorems that establish the existence of a fixed point, a point that is invariant under certain transformations. Such theorems establish the existence of such points, but generally do not indicate where that point can be found. Fixed-point theorems are important in many branches of mathematics and science.

One well-known fixed-point theorem is Brouwer's fixed-point theorem, which states that, for any function `f` that maps a compact convex set to itself, there must be some point `p` that is fixed under this transformation; in other words, it must be that, for some point `p`, `f`(`p`) = `p`. This theorem has some curious real-world applications. For example, if you stir a cup of coffee, there is always one point on the surface that is not in motion. Also, no matter how much you stir the coffee, there will always be one point in the coffee that will be in the same place after stirring as it was before. Also, if you have a piece of paper and put it in a box the same size as the piece of paper, then crumple or flatten the paper as much as you want (just don't tear the paper), and then put it back in the box, at least one point on the paper must be directly above the same spot in the box as it was before.

See also: The Wandering Monk.

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