Is there a generalization to Bolzano theorem when $f: \mathbb{R}^n \to \mathbb{R}^n$
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1$\begingroup$ The continuous image of a connected set is connected. $\endgroup$– Steven LandsburgCommented Apr 16, 2019 at 13:27
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3$\begingroup$ This site is for questions at research level. For general questions in mathematics see math.stackexchange.com. $\endgroup$– abxCommented Apr 16, 2019 at 13:46
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There can be various generalizations. My favorite one is this.
Let $f_1,\ldots,f_n$ be continuous functions on the unit cube. And suppose that each of them takes values of opposite sign on the opposite facets (each on its own pair of the opposite facets). Then all $f_j$ have a common zero inside the cube.
answered Apr 16, 2019 at 19:09
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2$\begingroup$ Is this Poincare Miranda theorem? $\endgroup$ Commented Apr 16, 2019 at 19:11
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$\begingroup$ Are there other generalizations? thank you $\endgroup$ Commented Apr 16, 2019 at 19:13
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1$\begingroup$ I am not sure about the name. It follows from the Brouwer theorem, see, for example Hurewicz and Wallman, Dimension theory, Ch. IV, 1D. $\endgroup$ Commented Apr 16, 2019 at 19:14
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$\begingroup$ Of course there are other generalizations. One of them stated in a comment to your question:-) $\endgroup$ Commented Apr 16, 2019 at 19:15
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1$\begingroup$ Yes, Wikipedia says that this is called Poincare-Miranda theorem. $\endgroup$ Commented Apr 17, 2019 at 0:10