Here are three fun Analysis questions, really aimed at my students who are studying their first/second course on Real Analysis, but hopefully it’s of wider interest as well.
We begin at the beginning, with a couple of definitions.
Definition. A function has the IVT Property if, for all real numbers with and there exists such that .
So, using this terminology, the Intermediate Value Theorem is the statement that a continuous function has the IVT Property.
Definition. Let and let . We say that is a weak limit of at if
for all there exists such that . Naturally, we say that has a weak limit somewhere if there exists such that is a weak-limit of at .
We say that is an accumulation point of a real sequence if for all there is an infinite set such that for all we have .
Show the existence or non-existence of a function which has the IVT Property but which is nowhere continuous.
Prove or disprove the claim that every function has a weak limit somewhere.
Find a real sequence such that every real number is an accumulation point of .
Where should we start? Perhaps my favourite of the three is the first one: to show the existence or non-existence of a function with the IVT property but which is nowhere continuous. There are several approaches we might take, beginning with the following lemma.
Lemma 1. Suppose that a function has the property that, for all with , the restriction of to the interval is surjective onto . Then has the IVT property.
Hopefully the proof of this lemma is quite clear: the IVT property asks, of such a function restricted to such an interval, that it attain certain values. So if attains all real values, then the property is evidently satisfied.
Okay…. So what?
Lemma 2. Suppose that a function has the property that, for all with , the restriction of to the interval is surjective onto . Then is continuous nowhere.
Proof. Let be given, and consider, for example, . For any , by assumption restricted to the open interval is surjective onto . In particular, the image under of is not a subset of .
Ah-ha! Now it’s clear why we’re interested in these sorts of function. Such a `locally surjective function‘ would be a nowhere continuous function which nevertheless has the IVT property. The big question now is: does any locally surjective function exist?
Well, YES. Such functions do exist, and there are several ways to find them. First we give a slightly abstract construction, which might seem a little unsatisfactory.
Method 1. Recall that the cardinality of is , and the cardinality of is also . We may choose a bijection
Also, let be the natural map sending each real number to its coset modulo , namely . Note that is surjective. Thus, the composition is also a surjection. Indeed, by the density of the rationals in the reals, each proper open interval contains a representative of each coset of in . That is, for each , there exists such that . A consequence of this is that the restriction to of is still surjective onto . Finally, the restriction of to is still surjective onto , which shows that is locally surjective, as required.
Wonderful, that was easy. But it wasn’t particularly constructive. Perhaps it felt a little unsatisfactory to rely on the cardinalities of these sets. Is there a better way?
That’s where I’ll leave it for the moment. To be continued!
In the meantime, try to construct a locally surjective function by using the decimal expansion of each real number.