Conference rusure::math

1917.0. "WANTED: First species discontinuity and related results" by EVTSG8::ESANU (Au temps pour moi) Wed Dec 14 1994 06:25

I am interested in results concerning the first species discontinuities.
Could you please contribute with the results that you know (bibliography
pointers welcome)?

Thank you.
Mihai.

--- --- ---

Here are some results:


Definition D1. A discontinuity point for a real function of a real variable
is of the first species if both side limits exist at that point.

Definition D2. A discontinuity point for a real function of a real variable
is of the second species if it is not of the first species.


These definitions can be naturally extended to functions defined on
linearly ordered topological spaces and taking values in arbitrary
topological spaces.


Let us note:
	Discont(f) = the set of the discontinuity points of the
	function f. 
	Discont(f,I) =  the set of the first species discontinuity points
	of the function f. 
	Discont(f,II) =  the set of the second species discontinuity points
	of the function f.

So we have   Discont(f) = Discont(f,I) U Discont(f,II)

Theorem T1 (I don't know its author - it's a classical theorem). If a real
function of a real variable is monotone, then its only discontinuity points
are of the first species.

Theorem T2 (Froda, 1929, [1]). The set of the first species discontinuity
points of a real function of a real variable is at most countable.

Theorem T3 (Yury Y. Prokhorov, 1956, [3]). If a function of a real variable
taking its values in a metric space has no second species discontinuity
points, then its set of discontinuity points is at most countable.

Theorem T4 (L.S. Gal, 1957, [4]). If a function of a real variable taking
its values in a metric space is right or left continuous in each point,
then its set of discontinuity points is at most countable.

Theorem T5 (Wang Yim-Ming, 1965, [5]). For a function of a real variable
taking its values in a metric space, the set of the points in which at
least one of the side limits exists is at most countable.


Definition D3. The weight of a topological space is defined as follows:
every set of cardinal numbers is well-ordered by  < , so that the set of
all cardinal numbers of the form  card(B) , where  B  is a base for the
topological space considered, has a smallest element; this cardinal number
is called the  weight of the topological space considered .

Definition D4. A topological space is  second countable  iff  its weight is
countable.


Theorem T6 (Your faithful, 1992, [6]). Let  X, Y  be separated (Hausdorff)
topological spaces, X linearly ordered and  f:X-->Y . Then the cardinal of
the set of the points  x in X  for which  f(x)  is not a left - or right -
cluster point of  f  is less than or equal to the product of the weights of
the spaces  X  and  Y .
                                                      
Corollary C6.1 (Your faithful, 1992, [6]). Let  X, Y  be separated
(Hausdorff) topological spaces, X linearly ordered and  f:X-->Y . Then the
cardinal of the set of the discontinuity points of  f  of the first species
is less than or equal to the product of the weights of the spaces  X  and 
Y .

Theorem T7 (Your faithful, 1992, [6]). Let  X, Y  be separated (Hausdorff)
topological spaces. If  f:X-->Y  has a limit in each point, then the
cardinal of the set of the discontinuity points of  f  is less than or
equal to the product of the weights of the spaces  X  and  Y .


Dan Preotescu ([7]) has generalized Froda's Theorem (T1), and the concept
of discontinuity points of the first species itself, to functions defined
on very general, *non-ordered!*, topological spaces. By the same token he
extended the results T6 and C6.1.


Bibliography:
------------

[1] Alexandru Froda, These de doctorat (Ph. D. Thesis, in French),
Gauthier-Villars, Paris, 1929

[2] Solomon Marcus, Din gindirea matematica romaneasca (From the Romanian
mathematical thinking, in Romanian), Editura Stiintifica si Enciclopedica,
Bucuresti, 1975

[3] Yu. Y. Prokhorov, Convergence of random processes and limit theorem in
probability, Teor. Veroyatnosti i Primenenie, I (1956), p. 177-238

[4] L.S. Gal, On the continuity and limiting values of functions, Trans.
American Math. Soc., 86(1957), p. 321-334

Theorem [5] Wang-Yim-Ming, A theorem on points of discontinuity of functions,
Journ. London Math. Soc., 40 (1965), p. 324-325

[6] Mihai Esanu, Une generalisation des theoremes de A. Froda et Yu. Y.
Prokhorov concernant les points de discontinuite de premiere espece (A
generalization of the theorems of A. Froda and Yu. Y. Prokhorov concerning
the first species discontinuity points, in French)
in 
Studii si Cercetari Matematice (Mathematical Reports), Tomul 4,
iulie-august 1992, Editura Academiei Romane, Bucuresti, p. 297-299

[7] Dan Preotescu, Private communication to Mihai Esanu, May 1993