TheoremProved

Intermediate value theorem

Statement

If $f:[a,b]\to\mathbb{R}$ is continuous and $y_0$ lies between $f(a)$ and $f(b)$ , then there exists $c\in[a,b]$ with $f(c)=y_0$ .

Why is it true?

A continuous path that starts below sea level and ends above sea level must cross sea level at some moment; a graph drawn without lifting the pen cannot jump over a height without passing through it.

Proof sketch

Assume without loss of generality $f(a)\le y_0\le f(b)$ . Let $c=\sup\{x\in[a,b]: f(x)\le y_0\}$ ; this set is nonempty (contains $a$ ) and bounded, so $c$ exists by completeness of $\mathbb{R}$ . Continuity of $f$ at $c$ forces $f(c)\le y_0$ (as a limit of points where $f\le y_0$ ) and $f(c)\ge y_0$ (otherwise nearby points to the right would still satisfy $f\le y_0$ , contradicting that $c$ is the supremum), so $f(c)=y_0$ .

Proved by

proved 1821Augustin-Louis Cauchy

Topics that use this theorem

Continuous Functions

Related theorems

Extreme value theorem Bolzano–Weierstrass theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Augustin-Louis Cauchy (1821). Cours d'analyse de l'École royale polytechnique
David M. Bressoud (2007). A Radical Approach to Real Analysis