论文标题
在所有线路上连续的功能不连续性集
On sets of discontinuities of functions continuous on all lines
论文作者
论文摘要
回答K.C.问的问题Ciesielski和T. Glatzer在2013年，我们在$ [0,1] $上构建了$ c^1 $ -Smooth函数$ f $ f $ in $ [0,1] $和A集成$ m \ subset \ subset \ operatotorname {graph} f $ nevere nhere nhere在$ \ operatatorNORNAME {graph} f $中都没有任何线性连续$ \ \ $^$^$^2 $^2 \ y MATHIN。I. $ m $的每个点不连续。我们实质上使用了最新的全面表征，即在2020年由T. Banakh和O. Maslyuchenko在$ \ Mathbb r^n $上线性连续功能的不连续点进行表征。这是我们结果的一个简单结果，我们证明，S.G. Slobobodnik在1976年证明了这种不满的条件。我们还证明了这种Slobodnik在可分开的Banach空间中的类似物。
Answering a question asked by K.C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a set $M \subset \operatorname{graph} f$ nowhere dense in $\operatorname{graph} f$ such that there does not exist any linearly continuous function on $\mathbb R^2$ (i.e. function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $\mathbb R^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S.G. Slobodnik in 1976 is not sufficient. We also prove an analogon of this Slobodnik's result in separable Banach spaces.
