## Symmetric semicontinuity implies continuity.(English)Zbl 0601.26003

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
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### References:

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