Intermediate Value Theorem

A rigorous, example-driven guide to intermediate value theorem, including hypotheses, method choice, verification, and practice.

Open the full lesson →

A limit describes the value approached from nearby inputs, whether or not the point is filled there. Left and right curve pieces approach the same open point while arrows show both directions of approach. A separate filled point illustrates that function value and limiting value are distinct questions.

The central idea

If $f$ is continuous on the closed interval $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then there exists at least one $c\in[a,b]$ such that $f(c)=N$. For root existence, take $N=0$: opposite signs at the endpoints guarantee a zero inside, provided continuity holds.

Definitions, hypotheses, and notation

Continuity on the entire closed interval is essential. A discontinuous function can have opposite endpoint signs while jumping over zero, so endpoint arithmetic alone is not enough. The target value may equal an endpoint output, in which case the theorem permits $c=a$ or $c=b$; strict between-ness places a solution in the open interval.

Bisection operationalizes the theorem. After finding an interval with opposite signs, evaluate the midpoint and retain the half whose endpoint values still bracket zero. Continuity preserves existence in every retained interval, while shrinking width controls location error. This algorithm still does not establish uniqueness; monotonicity or another argument must supply that.

A dependable method and decision rule

Identify a closed interval and the target output $N$.
Justify continuity on the entire interval, including any domain restrictions.
Evaluate the two endpoint values.
Show explicitly that $N$ lies between those values.
Conclude that at least one appropriate $c$ exists; do not claim uniqueness without more information.

Fully worked example

Common mistakes and why they fail

Existence is not a formula

On $[a,b]$, establish continuity throughout and show the target $L$ lies between $f(a)$ and $f(b)$. The theorem then guarantees some $c$ with $f(c)=L$, but does not locate it or guarantee uniqueness. For roots, take $L=0$ and bracket it with opposite signs. A sign change without continuity is insufficient: $1/x$ changes sign across zero and has no root. After proving existence, bisection can narrow the location by repeatedly keeping a subinterval whose endpoint outputs bracket the target. Each step preserves continuity on the smaller interval, which is why the numerical procedure rests on the theorem rather than on guesswork.