When does a map admit a lower “anti-continuity” bound like
for some function
and for all
? The answer is easy:
must be injective and its inverse must be uniformly continuous. End of story.
But recalling what happened with diameters of connected sets last time, let’s focus on the inequality for connected subsets
. If such
exists, the map f has the LOB property, for “lower oscillation bound” (oscillation being the diameter of image). The LOB property does not require
to be injective. On the real line,
satisfies it with
: since it simply folds the line, the worst that can happen to the diameter of an interval is to be halved. Similarly,
admits a lower oscillation bound
. This one decays faster than linear at 0, indicating some amount of squeezing going on. One may check that every polynomial has the LOB property as well.
On the other hand, the exponential function does not have the LOB property, since
tends to
as
. No surprise there; we know from the relation of continuity and uniform continuity that things like that happen on a non-compact domain.
Also, a function that is constant on some nontrivial connected set will obviously fail LOB. In topology, a mapping is called light if the preimage of every point is totally disconnected, which is exactly the same as not being constant on any nontrivial connected set. So, lightness is necessary for LOB, but not sufficient as shows.
Theorem 1: Every continuous light map with compact domain
admits a lower oscillation bound.
Proof. Suppose not. Then there exists and a sequence of connected subsets
such that
and
. We can assume
compact, otherwise replace it with its closure
which we can because
.
The space of nonempty compact subsets of is called the hyperspace of
; when equipped with the Hausdorff metric, it becomes a compact metric space itself. Pass to a convergent subsequence, still denoted
. Its limit
has diameter at least
, because diameter is a continuous function on the hyperspace. Finally, using the uniform continuity of
we get
, contradicting the lightness of
.

Here is another example to demonstrate the importance of compactness (not just boundedness) and continuity: on the domain define
. This is a homeomorphism, the inverse being
. Yet it fails LOB because the image of line segment
has diameter
, which can be arbitrarily close to 0. So, the lack of compactness hurts. Extending
to the closed square in a discontinuous way, say by letting it be the identity map on the boundary, we see that continuity is also needed, although it’s slightly non-intuitive that one needs continuity (essentially an upper oscillation bound) to estimate oscillation from below.
All that said, on a bounded interval of real line we need neither compactness nor continuity.
Theorem 2: If is a bounded interval, then every light map
admits a lower oscillation bound.
Proof. Following the proof of Theorem 1, consider a sequence of intervals such that
and
. There is no loss of generality in considering open intervals, since it can only make the diameter of the image smaller. Also WLOG, suppose
and
; this uses the boundedness of
. Consider a nontrivial closed interval
. For all sufficiently large
we have
, which implies
. Thus
is constant on
, a contradiction.
The property that distinguishes real line here is that nontrivial connected sets have nonempty interior. The same works on the circle and various tree-like spaces, but fails for spaces that don’t look one-dimensional.