# Continuous function/R/Intermediate value theorem/0-version/Fact

Zero value theorem

Let be real numbers, and let be a continuous function with and .

Then there exists an

such that

.
Zero value theorem

Let ${}a\leq b$ be real numbers, and let ${}f\colon [a,b]\rightarrow \mathbb {R}$ be a continuous function with ${}f(a)\leq 0$ and ${}f(b)\geq 0$.

Then there exists an

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

${}f(x)=0$.