# An Introduction to the Little Lip Function

Bruce Hanson, St. Olaf
PDL C-401

Given a continuous function \$f:\mathbb{R}^n \to\mathbb{R}\$ with \$M_f(x,r)=\frac{\sup_{|x-y|\le r}|f(x)-f(y)|}r\$, the so-called Big Lip" and Little Lip" functions are defined as follows:

\$Lip \ f(x)=\limsup_{r\to 0^+}M_f(x,r)\$         \$lip \ f(x)=\liminf_{r\to 0^+}M_f(x,r)\$

The Rademacher-Stepanov Theorem tells us that \$f\$ is differentiable almost everywhere on \$L_f=\{x \,|\,Lip \ f(x)<\infty\}\$. On the other hand, as Balogh and Csörnyei showed, this theorem no longer holds if we replace \$L_f\$ with \$l_f=\{x \,|\,lip\ f(x)<\infty\}\$. In this talk, I consider the problems of characterizing the sets \$E\subset\mathbb{R}\$ for which there exist a continuous function \$f\$ such that \$l_f=E\$ as well as characterizing the sets of non-differentiability for functions \$f\$ with \$l_f=\mathbb{R}\$. I will also examine some additional questions about the relationship between \$L_f\$ and \$l_f\$.

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