## Sur la quasi-continuité et la quasi-continuité approximative. (On quasi-continuity and approximate quasi-continuity).(French)Zbl 0657.26003

Let (R,$${\mathcal T})$$ be a topological space of the set of all reals. A function $$f:R\to R$$ is called $${\mathcal T}$$-quasi-continuous ($${\mathcal T}$$- cliquish) iff for every point $$x\in R,$$ every $${\mathcal T}$$-open set U containing x and every $$\epsilon >0$$ there exists a $${\mathcal T}$$-open set V such that $$V\subset U$$ and $$| f(t)-f(x)| <\epsilon$$ for each $$t\in V$$ $$(_{V} f<\epsilon).$$ Let $${\mathcal T}_ e({\mathcal T}_ d)$$ be the Euclidean (density) topology on R and Q, $$Q_ d$$, resp. (P, $$P_ d$$, resp.) be the set of all $${\mathcal T}_ e$$-quasi-continuous, resp. $${\mathcal T}_ d$$-quasicontinuous ($${\mathcal T}_ e$$-cliquish, resp. $${\mathcal T}_ d$$-cliquish) real functions of a real variable. If K is any family of real functions of a real variable, then we put $$B(K)=\{f:R\to R:$$ there exists a sequence $$\{f_ n\}^{\infty}_{n=1}$$ of K such that $$f(x)=\lim_{n\to \infty}f_ n(x)$$ for each $$x\in R\}.$$ Let D be the set of all $$f:R\to R$$ with the Baire property. There is proved: $$B(Q)=P,$$ $$B(P)=D,$$ $$B(Q_ d)=P_ d$$ and there exists a topology $${\mathcal T}$$ on R such that $$B(Q_{{\mathcal T}})\neq P_{{\mathcal T}},$$ where $$Q_{{\mathcal T}}(P_{{\mathcal T}})$$ is the set of all $${\mathcal T}$$-quasi-continuous ($${\mathcal T}$$-cliquish) functions.
Reviewer: L.Mišík

### MSC:

 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Full Text: