Symmetric semicontinuity implies continuity. (English) Zbl 0601.26003

Uher, Jaromir

Trans. Am. Math. Soc. 293, 421-429 (1986).

The author gives a very interesting view on the relation between the symmetric semicontinuity and the continuity. For this view the Lemma 1 is basic. We say that a function f is upper (lower) symmetrically semicontinuous at x iff \[ \limsup_{h\to 0+}(f(x+h)-f(x-h))\leq 0\quad (\liminf_{h\to 0+}(f(x+h)-f(x-h))\geq 0) \] and f is symmetrically semicontinuous at x iff it is upper or lower symmetrically semicontinuous at x. The main theorem is formulated by the generalized notion of family \({\mathcal T}=\{A\subset R: b(A)=\emptyset \}\) of exceptional sets, where b is a mapping of \(2^ R\) into itself satisfying the following nine conditions:
(i) If \(A\subset B\subset R\), then b(A)\(\subset b(B)\); (ii) If \(A=\cup^{\infty}_{n=1}A_ n\), then \(b(A)- \cup^{\infty}_{n=1}b(A_ n)\in {\mathcal T};\) (iii) If A-B\(\in {\mathcal T}\) and B-A\(\in {\mathcal T}\), then \(b(A)=b(B)\); \((iv)\quad b(b(A))=b(A);\) \((v)\quad A-b(A)\in {\mathcal T};\) \((vi)\quad b(\alpha +\beta A)=\alpha +\beta b(A)\) for each \(\alpha\),\(\beta\in R\). To formulate the last three conditions, it is useful to define that a set \(A\subset R\) is measurable iff b(A)-A\(\in {\mathcal T}\). (vii) Each in R open set is measurable. (viii) If A is measurable, then \(b(A)\cap b(B)=b(A\cap B).\) (ix) For any \(A_ 1,...,A_ k\), among them at least (k-1) are measurable and for which \(0\in b(A_ i)\) for all \(i=1,...,k\) and for any \(p>0\), there exists such a \(\Delta >0\) that \(I\cap \cap^{k}_{i=1}(\alpha_ i+A_ i)\not\in {\mathcal T}\) for any \(\delta\in (0,\Delta)\), for any open interval \(I\subset (-\delta,\delta)\) with length at least \(\delta\) /p and for any \(\alpha_ 1,...,\alpha_ k\in (-\delta,\delta).\)
We say that a property P holds at almost every point of a set A iff there exists such a \(B\in {\mathcal T}\) that P holds at every point of A-B.
The main theorem. Any real function f of a real variable is continuous at almost every point of the set R-b(\(\{\) \(x\in R:\) f is not symmetrically semicontinuous at \(x\})\).
There are also given two other theorems and a simple, but very nice example of D. Preiss related to the subject of the theorems.

Reviewer: L.Mišík

MSC:

26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Keywords:

connections between symmetric and ordinary differentiability; family of exceptional sets; relation between the symmetric semicontinuity and the continuity

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