One of the most direct ways to extend this lemma to n real dimensions involves the impossibility  of constructing a continuous locality function, given that a closed, connected region cannot be comprised of two disjoint domains.
One of the most direct ways to extend this lemma to n real dimensions involves the impossibility  of constructing a continuous locality function, given that a closed, connected region cannot be comprised of two disjoint domains. A modern version may be derived from the connectedness of a function from a connected topological space, given than intervals are the only connected subsets of the real numbers and, though Bolzano is often credited with the first statement of this theorem, his purported proof of it is erroneous. For ten points, name the fundamental mathematical theorem that is equivalent to Rolle's theorem and that was  stated by Weierstrass when he noted that a function that is somewhere positive and somewhere negative  must vanish at at least one point.
One of the most direct ways to extend this lemma to n real dimensions involves the impossibility  of constructing a continuous locality function, given that a closed, connected region cannot be comprised of two disjoint domains.
intermediate value theorem