Is there a lower semicontinuous function $f:[0,1] \to [0,1] $ such that the set $\{t\in (0,1] : \liminf_{s\nearrow t} f(s) > f(t)\}$ is uncountable?

Question

Let $f:[0,1] \to [0,1] $ be a lower semicontinuous function. I am interested in the set $$ S:= \left\{t\in (0,1] : \liminf_{s\nearrow t} f(s) > f(t)\right\}$$

Is there an example for which this is uncountable?

Note that $S$ is a subset of the points of discontinuity of $f$.

Motivation: I have some involved arguments in some other context (beginning with an arbitrary lower semicontinuous function), which suggest that this set is countable. I mistrusted this outcome and already checked for a weak link in the argumentation, which relies on the lower semicointinuity. But I did not find a weak link. Before I try to find something which probably isn't there I want to try to answer this question. A reference is also welcome.

What I've tried:

1. I tried to construct a function $f_C$ of the form $$f_C (t) := 1_{[0,1]\setminus C}(t),$$ where $C$ is a closed set (so the (fat) Cantorset is possible). But then $$S= \{ t \in (0,1]\cap C\; \vert\; \exists a < t : (a,t)\cap C = \emptyset \}$$ which is countable, because one can map it injectively into the rational numbers.

2. Without success I investigated $$S_\varepsilon := \left\{t\in (0,1] : \liminf_{s\nearrow t} f(s) \geq f(t)+ \varepsilon\right\}$$ because I suspected also a isolated-from-left structure.

Answer 1

Here is another proof:

Let $f:[0,1]\to \Bbb R$ be lower semicontinuous. If $$t_0 \in S_f := \left\{ t\in (0,1] : \liminf_{s\nearrow t} f(s) > f(t)\right\}$$ then there are $\varepsilon, \delta >0 $ such that $$ z := f(t_0) +\varepsilon \leq f(s) \quad \forall s \in (t_0-\delta , t_0).$$

Further, since $f$ is lower semicontinuous, we have that there is $x\in \Bbb R$ with $$x < y:=\inf_{s\in [0,t_0-\delta]} f(s).$$

Now, let $(B_t)_{t\geq 0 }$ be a standard Brownian motion and define $W_t := B_t +x$. Furthermore, define $$\tau_f := \inf\{t > 0: W_t \geq f(t)\},$$ which is measurable since $f$ is lower semicontinuous. Now observe that $$\Bbb P( \tau_f = t_0) \geq \Bbb P(\tau_y > t_0 -\delta , \tau_z > t_0 , W_t \in (f(t_0), z))\\ = \Bbb P \left(\sup_{s\in [0,t_0 -\delta]} (B_s +x) < y, \sup_{s\in [t_0 -\delta ,t]}(B_s +x) < z, B_{t_0} + x \in (f(t_0) , z)\right) >0.$$ This implies that $$S_f \subseteq \{t \geq 0 : \Bbb P (\tau_f = t_0) > 0\}.$$ But the set on the r.h.s. is countable.

If $f: [0,1] \to [0,1]$ is an arbitrary function, then $$f_* (t) := \min \left(\liminf_{s\to t}f(s) , f(t)\right)$$ is lower semicontinuous and $S_f \subseteq S_{f_*}$.

Answer 2

This answer only consists of the composed comments of Dave L. Renfro.

NO. The set S (indeed, even sets defined by much less restrictive conditions) is countable for ANY function (non-Borel, non-measurable, etc.) from an interval into an interval. See this mathoverflow answer.

The google books link I gave for $[1]$ (in the mathoverflow answer I cited) takes you directly to the page the 1907 paper begins. If you want a print copy, download the volume (not behind any pay-wall where I'm at) and print the pages for the article itself. Regarding books where it is mentioned or proved, this is not something I've tried collecting book references for, but many of the books given in this sci.math post should have the result, most likely references 7, 20, 22-25, 44.

Googling the title of the 1907 paper (as a phrase) in google-books will give some book references, but I suspect the vast majority of places where you can find a proof do not cite Young's paper, probably because many of Young's non-Fourier series results were ignored by the "wider mathematical community" after the 1910s and 1920s and, for historical issues, textbook authors rarely dig beyond what the "wider mathematical community" knows. I don't know the first proof of this result in a book:

It might be in Hausdorff's 1914 book; it's surely somewhere in Hahn's 1921 book--see Chapters II and III especially pp. 189-190--although I can't read German so I can't be more specific without spending a lot of time; it's definitely on pp. 287-288 of the 1921 2nd edition of Hobson's book.

However, the first mention of this result is almost certainly Note (α) on p. 64 of The Fundamental Theorems of the Differential Calclulus by Young (1910). Incidentally, Young's statement in his 1910 book only talks about where lim-inf differs from lim-sup, and not also where one or both of these differ from the value of the function at a point, and for some reason (doubtful it's because of this specific reference, since few know about it) many people (e.g. who cite Froda) overlook what Young actually proved in 1907.