Extreme and Intermediate Value Theorems

Continuity at a point is a local condition. On a closed bounded interval it yields two global theorems that ordinary calculus uses without proof: a continuous function on $[a, b]$ attains a highest and a lowest value, and it takes every height between its endpoint values. Both rest on the same property of $[a, b]$ (it is compact: closed and bounded at once), and both are proved from Bolzano–Weierstrass.

Boundedness

One proof template recurs below: build a sequence with a wanted property, then use Bolzano–Weierstrass to make it converge, with closedness returning the limit to the interval.

The extreme value theorem

On a closed bounded interval a continuous graph reaches a highest point at c and a lowest at d; both x-locations lie inside the interval.

Each hypothesis is necessary; dropping any one makes the conclusion fail.

Failure	Example	What breaks
Interval unbounded	$f (x) = x$ on $R$	no Bolzano–Weierstrass; no max or min
Interval not closed	$f (x) = 1/ x$ on $(0, 1)$	limit of subsequence escapes the domain
$f$ not continuous	$f (x) = 1/ x$ ( $x > 0$ ), $f (0) = 0$ on $[0, 1]$	continuity fails at $0$ ; no max

Stated topologically, the theorem is one line: the continuous image of a compact set is compact, hence closed and bounded, hence contains its own supremum and infimum.

The intermediate value theorem

The second theorem says a continuous function skips no values between its endpoint heights. The proof rests on a bisection lemma that doubles as a numerical algorithm.

Bisection brackets a root: at each step the half whose endpoints still straddle zero is kept, and the nested brackets shrink to the crossing point c.

For any target height y between the endpoint values, the graph must cross the level line y = target, giving a point c with f(c) = y.

Stated topologically, IVT is that the continuous image of a connected set is connected: $f ([a, b])$ is an interval, and an interval containing $f (a)$ and $f (b)$ contains everything between. Sharpening the boundedness lemma and IVT together yields that the direct image $f ([a, b])$ is itself a closed bounded interval (or a single point).

Applications

Bisection as an algorithm

The proof of the sign-change lemma is a root-finder. It converges for any continuous function, needs only sign evaluations, and reaches any target precision in finitely many steps.

Algorithm:

\textsc{Bisect}(f, a, b, \tau)

— bracket a root of a continuous

f

with

f(a)\,f(b) < 0

1
while $b - a > \tau$ do
2
$m \gets (a + b) / 2$
3
if $f(m) = 0$ then
4
return $m$
5
else if $f(a) \cdot f(m) < 0$ then
6
$b \gets m$
root lies in the left half
7
else
8
$a \gets m$
root lies in the right half
9
end if
10
end while
11
return $(a + b) / 2$

Faster methods exist, Newton's among them, but bisection's guarantee is unconditional once a sign change is found.

Existence proofs

Odd-degree polynomials have a real root. A monic odd-degree polynomial $g$ satisfies $g (x) \to \infty$ as $x \to \infty$ and $g (x) \to - \infty$ as $x \to - \infty$ , so $g$ is positive at some large $M$ and negative at some large $- K$ ; IVT gives a root in $(- K, M)$ .
$k$ -th roots exist. For $y > 0$ and $k \in N$ , the polynomial $f (x) = x^{k} - y$ is negative at $0$ and positive far enough right, so IVT gives an $x > 0$ with $x^{k} = y$ .
Fixed points. Any continuous $f : [0, 1] \to [0, 1]$ has a fixed point: apply IVT to $g (x) = f (x) - x$ , which is $\geq 0$ at $0$ and $\leq 0$ at $1$ .

An odd-degree monic polynomial runs from below the axis at large negative inputs to above it at large positive inputs, so it must cross zero somewhere between: a guaranteed real root.

Two of these existence facts are worth spelling out as concrete computations.

A continuous self-map of an interval must meet the diagonal y = x: the graph starts on or above it and ends on or below it, so it crosses, and the crossing is a fixed point.

The intermediate value property does not, by itself, imply continuity: $sin (1/ x)$ extended by $0$ at the origin has the IVT property while being discontinuous there. A stronger condition, a single $δ$ that works uniformly across the whole interval, is what a continuous function on $[a, b]$ automatically satisfies: uniform continuity.

Lebl, Basic Analysis I, §3.3 — Lemma 3.3.1 (a continuous function on a closed bounded interval is bounded). ↩
Lebl, Basic Analysis I, §3.3 — Theorem 3.3.2 (minimum–maximum / extreme value theorem) and the examples showing the hypotheses are needed. ↩
Lebl, Basic Analysis I, §3.3 — Lemma 3.3.7, Theorem 3.3.8 (Bolzano's intermediate value theorem), Proposition 3.3.10, and Corollary 3.3.13. ↩ ↩²

Boundedness

The extreme value theorem

The intermediate value theorem

Applications

Bisection as an algorithm

Existence proofs

Footnotes