References on semicontinuous functions

Question

The generalisation of continuity to semicontinuity is well-known. I suppose it should be also well-studied. The only references I found offhand are the ones from the wikipedia entry semi-continuity. After some time I also found the book "Reelle Funktionen" (1921) by Hans Hahn together with the article "Über halbstetige Funktionen und deren Verallgemeinerung" (1919) by Felix Hausdorff.

Edit: In particular, I search for results which concern the relation of the behaviour of lowersemicontinuous functions on a dense subset and their behaviour on the whole domain. But additionally I want to get more into the topic as well.

Answer 1

A classical source is Hobson "The theory of functions of a real variable vol 1" free available here. This is a quite old text but, to me, is a classical reference. You can find semicontinuity at pag 237-240.

Answer 2

The following three books are good places to begin. I may add more later, but my criteria will be to only include items that are in English (I might later decide to deviate from this) and which have a more-than-usual amount about semicontinuous functions.

[1] Eduard Čech, Point Sets, translation by Ale Pultr of the 1966 Czech edition, Academia, Publishing House of the Czechoslovak Academy of Sciences, 1969, 271 pages.

Section 14: Functions of the first class (pp. 78-91) is mostly concerned with Baire 1 functions (a class of functions that properly includes all semicontinuous functions), but there are some useful results on semicontinuous functions, and there is a lot of information that is relevant to semicontinuous functions.

[2] Isidor [Isidore] Pavlovich Natanson, Theory of Functions of a Real Variable, Volume II, translated by Leo Francis Boron from the 1957 Russian edition, Frederick Ungar Publishing Company, 1960, 265 pages. Dover edition

See Chapter XV: The Baire Classification (pp. 128-156), especially Section 3: Functions of the First Class (pp. 139-149) and Section 4: Semi-continuous Functions (pp. 149-156). Note that Natanson's definition of the $\limsup$ and $\liminf$ of a function at a point differs from the standard usage (see my comment here for details), and thus his definition of lower semicontinuous and upper semicontinuous looks different (but isn't different) from what you'll see in other books. Nonetheless, Natanson is a great reference for carefully proved results and useful examples.

[3] Arnaud Casper Maria van Rooij and Wilhelmus Hendricus Schikhof, A Second Course on Real Functions, Cambridge University Press, 1982, xiv + 200 pages.

Section 10: Semicontinuous functions (pp. 59-65) contains a large number of results (most as exercises for the reader, some with hints). This is followed by Section 11: Functions of the first class of Baire (pp. 65-74) that is also recommended for anyone interested in semicontinuous functions.

Answer 3

I will also list the references I found apart from the other answers.

[1] Ryszard Engelking, [General Topology] Translated from the Polish by the author. Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. viii+529 pp. ISBN: 3-88538-006-4.

Semicontinuous functions I (p.61-63) give a overwiew over many general results and properties (also hahn's "insertion" lemma). Semicontinuous function II (p.113) give results related to normal spaces (including specified hahn's "insertion" lemma). Problem 3.12.23. (g) (p.242) states an result related to countable compactness. Semicontinuous functions III (p.347) specifies hahn's "insertion" lemma to normal and countable paracompact spaces

[2] Georg Aumann, [Reelle Funktionen] Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete; Bd. 68 (alle Bände), 2. Aufl., Berlin [u.a.] : Springer, 1969

5.4 Halbstetige Funktionen (p.150-159) contains the "standard" results; is listed as reference of Rational Extensions of C(X) and Semicontinuous Functions by Jürg Schmid (1988) from the comments; Note that he defines the notions of $\liminf$ and $\limsup$ the same way as in Natanson, Theory of Functions of a Real Variable pointed out in the answer of Dave L. Renfro

[3] Łojasiewicz, Stanisław, [An introduction to the theory of real functions] A Wiley-Interscience Publication, Chichester [u.a.] : Wiley, 1988

3.3 Semicontinuous Functions, 3.4 Maximum and Minimum at a Point, 3.5 Functions of the first class of Baire (p.51-64) contains also the basic results; the presentation seems a bit sloppy to be

[4] Ene, Vasile, [Real functions : current topics] Lecture notes in mathematics; 1603 (alle Bände), Berlin [u.a.] : Springer, 1995

1.15 Semicontinuity; $\mathcal{S}$-semicontinuity (p.21-23) presents a generalisation of semicontinuity (in terms of a generalisation of open sets), some characterisations, but nothing more; *6.32 A lower semicontinuous Function $F\in AC_2 , \notin \underline{AC}$ * (p.244) states that the indicator function of the Cantor set is lowersemicontinuous but not $\underline{AC}$ (which seems to me to be a sort of lower absolute continuity)

For completeness I will also list the following:

Mikolás, Miklós; Real functions, abstract spaces and orthogonal series; Budapest : Akadémiai Kiadó; 1994

3.3 Properties of continuous and semi-continuous functions, (p.80-83) contains next to nothing but a definition and even the definition seems unusual to me, and the presentation is awful; he defines a function $f$ to be lsc at $x_0$ iff $\liminf_{x\to x_0, x\neq x_0} f(x) = f(x_0)$, which is actually stronger than the usual definition