This thesis deals with the generalization of a well-known theorem in one dimension, namely the Intermediate value theorem. Specifically, its corollary, which proves that a function has a root.
This theorem is well known. Students learn about it already in high school. For math students, this is almost a self-evident result. The theorem says that every continuous function on a closed interval that changes sign at the boundary points has at least one root in this interval.
With some modifications, this theorem can be generalized to an arbitrary dimension.
We prove that every continuous map on a unit cube in a $n$-dimensional Euclidean space has, under a certain condition, at least one root. The assumption we need is that each component function of the map changes sign on the corresponding sides of the unit cube.
Here, we also show the equivalence of this generalization of a one-dimensional theorem, namely the Poincaré-Miranda theorem, and The Brouwer fixed point theorem. We give a discrete proof of the Poincaré-Miranda theorem using the Steinhaus chessboard theorem.
We present possible generalizations of the Poincaré-Miranda theorem to certain infinite-dimensional spaces.
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