Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,m_s$ with $v(t)=m_i$ whenever $t \in I_i := [t_{i-1},t_i)$ for $i=1,\dots,s$. Let $\mathcal{P}$ denote the collection of intervals $I_i$ that make up the partition of [a,b].

Now, let $u: [a,b] \rightarrow \mathbb{R}$ be a regulated function (that is, $u$ is the uniform limit of a sequence of piecewise constant functions $u_n: [a,b] \rightarrow \mathbb{R}$) each $u_n$ of which is defined via a corresponding partition $\mathcal{P}_n$ of $[a,b]$ as described above.

Does anyone know of a condition that characterizes when such a function $u$ is $\alpha$-Hölder continuous for some $0<\alpha<1$?

E(up)lio.