On a closed interval of $\mathbb{Q}$ there is a continuous unbounded function, a bounded one with no maximum, and one without the intermediate value property

The notion of continuity used here is stated in full, and is not imported. Let $F$ be an ordered field, $D \subseteq F$ and $\varphi : D \to F$. Say $\varphi$ is continuous at $c \in D$ when

$\text{for every } \varepsilon > 0 \text{ in } F \text{ there is } \delta > 0 \text{ in } F \text{ with } |\varphi(x) - \varphi(c)| < \varepsilon \text{ whenever } x \in D \text{ and } |x - c| < \delta,$

and continuous on $D$ when it is continuous at every point of $D$. This is the ordinary $\varepsilon$-$\delta$ condition, read entirely inside $F$. Nothing below cites a definition of continuity from elsewhere in this library, because there is none yet.

Refuted claim: over every ordered field $F$, a function $\varphi : [a,b]_F \to F$ that is continuous on the closed interval $[a,b]_F$ (Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field) is bounded there, attains a maximum there, and takes every value between $\varphi(a)$ and $\varphi(b)$. In other words, the extreme value theorem and the intermediate value theorem hold over an arbitrary ordered field.

The witness is $F = \mathbb{Q}$ and $D = [0,2]_{\mathbb{Q}}$, with three functions, one for each clause:

$f(x) = \frac{1}{x^2 - 2}, \qquad g(x) = -\,(x^2-2)^2, \qquad h(x) = x^2 - 2 .$

All three are continuous on $D$ in the sense above. $f$ is unbounded; $g$ is bounded and has no maximum; $h$ satisfies $h(0) = -2 < 0 < 2 = h(2)$ and never takes the value $0$. What $\mathbb{Q}$ lacks is the least-upper-bound property (LUB) of The five completeness properties of an ordered field: least upper bound, monotone convergence, nested intervals, Bolzano-Weierstrass, and Cauchy completeness, and each of the three clauses fails because of that single omission.

• One mechanism, three failures. All three functions are built from $x^2 - 2$, whose zero is missing from $\mathbb{Q}$. The map $T$ is a contraction towards that missing zero: it halves $|x^2-2|$ at every step while staying inside $D$. So $|x^2-2|$ has infimum $0$ on $D$ and does not attain it, and the three failures are three ways of reading that one sentence.

• Nothing here is peculiar to $\mathbb{Q}$. The same construction runs in any ordered subfield of $\mathbb{R}$ that omits $\sqrt 2$, since every step above uses only the field operations, the order, and the absence of a square root of $2$. This item exhibits the cheapest witness; no claim is made here about ordered fields in general.

• This item does not use, and does not need, a general theory of continuous functions. The $\varepsilon$-$\delta$ condition is stated in the Statement refuted and every use of it above is a direct verification, so the item is self-contained and nothing here waits on a later page. That is deliberate and not a placeholder: the claim refuted here is a claim about an arbitrary ordered field, and it is refuted over $\mathbb{Q}$, so a definition of continuity written for real functions on subsets of $\mathbb{R}$ would not apply to it. This library has no notion of continuity over a general ordered field and needs none elsewhere, and inventing an id for one would put an unused definition on a page about completeness properties. The condition above is the ordinary one read inside $F$, and it specialises to the real-variable definition at $F = \mathbb{R}$.

• What is true over $\mathbb{Q}$. Continuity, sums and products of continuous functions, and composition all behave normally; what fails is every statement whose proof needs a supremum. That is the content of the page this one belongs to.