# Continuous function/R/Intermediate value theorem/Fact

Intermediate value theorem

Let be real numbers, and let be a continuous function. Let be a real number between and .

Then there exists an

such that

.
Intermediate value theorem

Let ${}a\leq b$ be real numbers, and let ${}f\colon [a,b]\rightarrow \mathbb {R}$ be a continuous function. Let ${}u\in \mathbb {R}$ be a real number between ${}f(a)$ and ${}f(b)$.

Then there exists an

${}x\in [a,b]$ such that

${}f(x)=u$.