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Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous mapping on the space of Dedekind real numbers, and let $c:p_{*}(\mathbb{R})$ be a term of the underlying homotopy type of the Dedekind real numbers. If we assume excluded middle, axiom R-flat, and the analytic Markov's principle, then if there exist points $a, b:\mathbb{R}$ such that $p_*(f)(p_*(a)) \lt c \lt p_*(f)(p_*(b))$, then there exists a point $x:\mathbb{R}$ such that $p_*(f)(p_*(x)) = c$
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous mapping on the Dedekind real numbers, and let $c:\mathbb{R}$ be a Dedekind real number. If we assume axiom R-flat, then if there exist points $a, b:\mathbb{R}$ such that $f(a)) \lt c \lt f(b))$, then for all terms $\epsilon \gt 0$ there exists a point $x:\mathbb{R}$ such that $\vert f(x)) - c \vert \lt \epsilon$
Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)
Matthew Frank, Interpolating Between Choices for the Approximate Intermediate Value Theorem, Logical Methods in Computer Science Vol 16 (3) (July 14, 2020) (arXiv:1701.02227, doi=10.23638/LMCS-16(3:5)20202020))
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