论文标题
连续函数具有不可渗透图
Continuous functions with impermeable graphs
论文作者
论文摘要
我们在单位间隔上构建了Hölder连续函数，该函数在不计其数（实际上是连续性）中重合的，每个函数的总变异的每个函数都小于1通过该来源。 我们说，具有此属性的功能具有不可渗透的图，并且我们提供了具有可渗透和不可渗透图的功能的进一步示例。 随后，第一个示例函数用于构建平面上连续函数的示例，该函数本质上是Lipschitz在hölder连续函数的互补的互补上连续的，但它并不是平面上的Lipschitz连续的Lipschitz。 作为另一个主要结果，我们在单位间隔上构建一个连续函数，该函数在一组Hausdorff维度1中重合，每个函数的每个函数都小于1的总变化。
We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with this property has impermeable graph, and we present further examples of functions both with permeable and impermeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.
