Intermediate value theorem

Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous. Let $y$ be a real number between $f(a)$ and $f(b)$. Then there is $c \in [a, b]$ with $f(c) = y$.

Remarks

Assume $f(a) < y < f(b)$. Let $c$ be the supremum of $\{ x \in [a, b] : f(x) < y \}$. Continuity gives $f(c) \le y$ and $f(c) \ge y$. The proof needs the completeness of $\mathbb{R}$. Over $\mathbb{Q}$, the function $x^2 - 2$ changes sign on $[1, 2]$ but has no rational zero. It gives the existence of $n$-th roots, and it shows that each odd degree real polynomial has a real root. Mathlib states it as intermediate_value_Icc.

History

Bernard Bolzano first proved the theorem in 1817.

Explanations

L1 human

A continuous function on an interval cannot skip values. To get from $f(a)$ to $f(b)$, its graph must cross every horizontal line in between.