The intermediate value theorem

A continuous function cannot pass from one value to another by “skipping” the intermediate values: it must take them all.

Theorem — The intermediate value theorem

If $f$ is continuous on $[a,b]$ and $k$ is a value lying between $f(a)$ and $f(b)$, then there exists at least one $c\in(a,b)$ such that $f(c)=k.$

Remark — Meaning

“A continuous function cannot jump from one value to another without passing through all the intermediate values.” It is the mathematical analogue of the physical fact that a moving object cannot teleport: to go from one height to another it must cross all the heights in between.

Links

Topics: Continuita
Concepts: Continuita · Teorema dei valori intermedi