On the extension of some functions to $Q_s1$-functions and on the sums of two $Q_s1$-functions

Rok, strany: 2006, 167 - 172

A function $f:Bbb R^m o Bbb R$ satisfies the condition $Q_s1$ at a point ${oldkey x}in Bbb R^m$ if for each real $ε > 0$ and for each set $U i {oldkey x}$ belonging to the density topology there is an open set $V$ such that $emptyset eq U cap V subset f^-1((f({oldkey x})-ε, f({oldkey x}) +ε)) cap C(f)$, where $C(f)$ denotes the set of all continuity points of $f$. For a nonempty set $Asubset Bbb R^m$ it is proved that the Lebesgue measure $łambda (cl(A)ig)=0$ if and only if for each $łambda$-almost everywhere continuous function $f:Bbb R^m o Bbb R$ there is a function $gin Q_s1$ such that $f|_A = g|_A$. Moreover, it is proved that every function $f:Bbb R^m o Bbb R$ satisfying the condition $łambda (cl(D(f)))=0,$ where $D(f)=Bbb R^m setminus C(f)$, is the sum of two functions $g,h:Bbb R^m o Bbb R$ with the condition $Q_s1$.

ISO 690:

Strońska, E. 2006. On the extension of some functions to $Q_s1$-functions and on the sums of two $Q_s1$-functions. In Tatra Mountains Mathematical Publications, vol. 34, no.2, pp. 167-172. 1210-3195.

APA:

Strońska, E. (2006). On the extension of some functions to $Q_s1$-functions and on the sums of two $Q_s1$-functions. Tatra Mountains Mathematical Publications, 34(2), 167-172. 1210-3195.