Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Solution 1:

I.

Some of the answers reveal a confusion, so let me start with the definition. If $I$ is an interval, and $f:I\to\mathbb R$, we say that $f$ has the intermediate value property iff whenever $a<b$ are points of $I$, if $c$ is between $f(a)$ and $f(b)$, then there is a $d$ between $a$ and $b$ with $f(d)=c$.

If $I=[\alpha,\beta]$, this is significantly stronger than asking that $f$ take all values between $f(\alpha)$ and $f(\beta)$:

• For example, this implies that if $J\subseteq I$ is an interval, then $f(J)$ is also an interval (perhaps unbounded).
• It also implies that $f$ cannot have jump discontinuities: For instance, if $\lim_{x\to t^-}f(x)$ exists and is strictly smaller than $f(t)$, then for $x$ sufficiently close to $t$ and smaller than $t$, and for $u$ sufficiently close to $f(t)$ and smaller than $f(t)$, $f$ does not take the value $u$ in $(x,t)$, in spite of the fact that $f(x)<u<f(t)$. This indicates that if $f$ is discontinuous, its discontinuities must be somewhat wild.
• In particular, if $f$ is discontinuous at a point $a$, then there are $y$ such that the equation $f(x)=y$ has infinitely many solutions near $a$.

II.

There is a nice survey containing detailed proofs of several examples of functions that both are discontinuous and have the intermediate value property: I. Halperin, Discontinuous functions with the Darboux property, Can. Math. Bull., 2 (2), (May 1959), 111-118. It contains the amusing quote

Until the work of Darboux in 1875 some mathematicians believed that [the intermediate value] property actually implied continuity of $f(x)$.

This claim is repeated in other places. For example, here one reads

In the 19th century some mathematicians believed that [the intermediate value] property is equivalent to continuity.

This is very similar to what we find in A. Bruckner, Differentiation of real functions, AMS, 1994. In page 5 we read

This property was believed, by some 19th century mathematicians, to be equivalent to the property of continuity.

Though I have been unable to find a source expressing this belief, that this was indeed the case is supported by the following two quotes from Gaston Darboux's Mémoire sur les fonctions discontinues, Ann. Sci. Scuola Norm. Sup., 4, (1875), 161–248. First, on pp. 58-59 we read:

Au risque d'être trop long, j'ai tenu avant tout, sans y réussir peutêtre, à être rigoureux. Bien des points, qu'on regarderait à bon droit comme évidents ou que l'on accorderait dans les applications de la science aux fonctions usuelles, doivent être soumis à une critique rigoureuse dans l'exposé des propositions relatives aux fonctions les plus générales. Par exemple, on verra qu'il existe des fonctions continues qui ne sont ni croissantes ni décroissantes dans aucun intervalle, qu'il y a des fonctions discontinues qui ne peuvent varier d'une valeur à une autre sans passer par toutes les valeurs intermédiaires.

The proof that derivatives have the intermediate value property comes later, starting on page 109, where we read:

En partant de la remarque précédente, nous allons montrer qu'il existe des fonctions discontinues qui jouissent d'une propriété que l'on regarde quelquefois comme le caractère distinctif des fonctions continues, celle de ne pouvoir varier d'une valeur à une autre sans passer par toutes les valeurs intermediaires.

III.

Additional natural assumptions on a function with the intermediate value property imply continuity. For example, injectivity or monotonicity.

Derivatives have the intermediate value property (see here), but there are discontinuous derivatives: Let $$f(x)=\left\{\begin{array}{cl}x^2\sin(1/x)&\mbox{ if }x\ne0,\\0&\mbox{ if }x=0.\end{array}\right.$$ (The example goes back to Darboux himself.) This function is differentiable, its derivative at $0$ is $0$, and $f'(x)=2x\sin(1/x)-\cos(1/x)$ if $x\ne0$, so $f'$ is discontinuous at $0$.

This example allows us to find functions with the intermediate value property that are not derivatives: Consider first $$g(x)=\left\{\begin{array}{cl}\cos(1/x)&\mbox{ if }x\ne0,\\ 0&\mbox{ if }x=0.\end{array}\right.$$ This function (clearly) has the intermediate value property and indeed it is a derivative, because, with the $f$ from the previous paragraph, if $$h(x)=\left\{\begin{array}{cl}2x\sin(1/x)&\mbox{ if }x\ne 0,\\ 0&\mbox{ if }x=0,\end{array}\right.$$ then $h$ is continuous, and $g(x)=h(x)-f'(x)$ for all $x$. But continuous functions are derivatives, so $g$ is also a derivative. Now take $$j(x)=\left\{\begin{array}{cl}\cos(1/x)&\mbox{ if }x\ne0,\\ 1&\mbox{ if }x=0.\end{array}\right.$$ This function still has the intermediate value property, but $j$ is not a derivative. Otherwise, $j-g$ would also be a derivative, but $j-g$ does not have the intermediate value property (it has a jump discontinuity at $0$). For an extension of this theme, see here.

In fact, a function with the intermediate value property can be extremely chaotic. Katznelson and Stromberg (Everywhere differentiable, nowhere monotone, functions, The American Mathematical Monthly, 81, (1974), 349-353) give an example of a differentiable function $f:\mathbb R\to\mathbb R$ whose derivative satisfies that each of the three sets $\{x\mid f'(x)>0\}$, $\{x\mid f'(x)=0\}$, and $\{x\mid f'(x)<0\}$ is dense (they can even ensure that $\{x\mid f'(x)=0\}=\mathbb Q$); this implies that $f'$ is highly discontinuous. Even though their function satisfies $|f'(x)|\le 1$ for all $x$, $f'$ is not (Riemann) integrable over any interval.

On the other hand, derivatives must be continuous somewhere (in fact, on a dense set), see this answer.

Conway's base 13 function is even more dramatic: It has the property that $f(I)=\mathbb R$ for all intervals $I$. This implies that this function is discontinuous everywhere. Other examples are discussed in this answer.

Halperin's paper mentioned above includes examples with even stronger discontinuity properties. For instance, there is a function $f:\mathbb R\to\mathbb R$ that not only maps each interval onto $\mathbb R$ but, in fact, takes each value $|\mathbb R|$-many times on each uncountable closed set. To build this example, one needs a bit of set theory: Use transfinite recursion, starting with enumerations $(r_\alpha\mid\alpha<\mathfrak c)$ of $\mathbb R$ and $(P_\alpha\mid\alpha<\mathfrak c)$ of its perfect subsets, ensuring that each perfect set is listed $\mathfrak c$ many times. Now recursively select at stage $\alpha<\mathfrak c$, the first real according to the enumeration that belongs to $P_\alpha$ and has not been selected yet. After doing this, continuum many reals have been chosen from each perfect set $P$. List them in a double array, as $(s_{P,\alpha,\beta}\mid\alpha,\beta<\mathfrak c)$, and set $f(s_{P,\alpha,\beta})=r_\alpha$ (letting $f(x)$ be arbitrary for those $x$ not of the form $s_{P,\alpha,\beta}$).

To search for references: The intermediate value property is sometimes called the Darboux property or, even, one says that a function with this property is Darboux continuous.

An excellent book discussing these matters is A.C.M. van Rooij, and W.H. Schikhof, A second course on real functions, Cambridge University Press, 1982.

Solution 2:

Let $f(x) = x^2\sin(1/x)$ on $(0,1)$ and $f(0) = 0$. Then $f$ is differentiable throughout $[0,1]$. All derivatives satisfy the intermediate value property (Darboux's Theorem); but $f'(x)$ is discontinuous at $0$.