The intermediate value property (Darboux property) of a function on an interval: the image of every subinterval is order-convex

Let $I \subseteq \mathbb{R}$ be order-convex (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f : I \to \mathbb{R}$. Then $f$ has the intermediate value property, also called the Darboux property, when

$f[J] \ \text{is order-convex for every order-convex } J \subseteq I .$

As everywhere in this library, "interval" is read as "order-convex" (A subset of $\mathbb{R}$ is connected if and only if it is order-convex, that is, an interval is what licenses the word; Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length records that the classification of the order-convex subsets of $\mathbb{R}$ into the nine written forms is not proved here).

The equivalent pointwise form

$f$ has the intermediate value property if and only if

for all $a, b \in I$ with $a < b$ and every real $y$ with $f(a) \le y \le f(b)$ or $f(b) \le y \le f(a)$, there is $c \in [a,b]$ with $f(c) = y$.

From the displayed condition to the pointwise one. Given $a < b$ in $I$, the set $[a,b]$ is order-convex and contained in $I$ by order-convexity of $I$, so $f[\,[a,b]\,]$ is order-convex; it contains $f(a)$ and $f(b)$, hence every $y$ between them, and such a $y$ is $f(c)$ for some $c \in [a,b]$.

From the pointwise condition to the displayed one. Let $J \subseteq I$ be order-convex, let $u, v \in f[J]$ and let $u \le y \le v$. Write $u = f(a)$ and $v = f(b)$ with $a, b \in J$. If $a = b$ then $u = v = y$ and $y \in f[J]$. If $a < b$, the pointwise condition gives $c \in [a,b]$ with $f(c) = y$, and $c \in J$ because $J$ is order-convex and $a, b \in J$; so $y \in f[J]$. If $b < a$ the same argument applies with the roles of $a$ and $b$ exchanged, the pointwise condition being stated symmetrically in the two orders. Hence $f[J]$ is order-convex.

Both forms are used below, and they are used interchangeably.

Every continuous function on an interval has the property

If $f$ is continuous on $I$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point) then $f[J]$ is order-convex for every order-convex $J \subseteq I$ (The image of an interval under a continuous real function is order-convex, hence an interval, and the image of a closed bounded interval is a closed bounded interval, claim 1). So continuity implies the intermediate value property.

The converse is false, and that is the whole reason the property is given a name of its own: a function may take every intermediate value on every subinterval and be continuous nowhere. The failure is recorded as FALSE: a function with the intermediate value property on an interval is continuous.

A monotone function with the intermediate value property is continuous. This is not a further theorem but a reading of A function on an interval satisfying $f(x) \le f(y)$ whenever $x \le y$, whose image is order-convex, is continuous: for a function satisfying $f(x) \le f(y)$ whenever $x \le y$ on an order-convex $I$, order-convexity of the single image $f[I]$ already forces continuity. So the pathologies live entirely among the non-monotone functions.