# Homotopy Type Theory intermediate value theorem > history (changes)

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## Definition

### Discontinuous intermediate value theorem

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 $p_*(f)(p_*(x)) = c$

### Approximate intermediate value theorem

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 $\vert f(x)) - c \vert \lt \epsilon$