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

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

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 $\epsilon$-$\delta$ 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.