关于位置的无处可微函数
Nowhere differentiable functions with respect to the position
摘要：
设$Ω$是$\mathbb{C}$中的一个有界域，使得$\partial Ω$不包含孤立点。设$R(Ω)$是$\overlineΩ$上极点偏离$\overlineΩ$的有理函数的一致极限空间，赋上确界范数.我们证明，要么$R(Ω)$中的所有函数$f$一般都满足每个$z_0 \in \partial Ω$的%$$ \limsup_{\substack{z \to z_0 z \in \partial Ω}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$，要么$R(Ω)$中没有此类函数满足此要求。在第一种情况下，通用函数$f \in R(Ω)$在$\partial Ω$上关于位置是无处可微的。我们给出了具体的例子，在每个情况下，前面的二分法举行。我们还将前面的结果推广到无界域。
英文摘要：
Let $Ω$ be a bounded domain in $\mathbb{C}$ such that $\partial Ω$ does not contain isolated points. Let $R(Ω)$ be the space of uniform limits on $\overlineΩ$ of rational functions with poles off $\overlineΩ$, endowed with the supremum norm. We prove that either generically all functions $f$ in $R(Ω)$ satisfy % $$ \limsup_{\substack{z \to z_0 z \in \partial Ω}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$ for every $z_0 \in \partial Ω$ or no such function in $R(Ω)$ meets this requirement. In the first case, the generic function $f \in R(Ω)$ is nowhere differentiable on $\partial Ω$ with respect to the position. We give specific examples where each case of the previous dichotomy holds. We also extend the previous result to unbounded domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
