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.



Hölder continuity of uniform limit of piecewise constant functions
I can't think of a characterization which is not too close to a tautology; a sufficient condition is the following. Denote the modulus of the subdivision $\mathcal{P}$ by $\|\mathcal{P}\|:=\max \_ {1\le i\le s} (t \_ i-t \_ {i-1})$, and by $\mathcal{P}^M$ the set of mid-points of the intervals $I\in \mathcal{P}$.


Assume that


**1.** $\|\mathcal{P \_ n}\|\to0\, ;$ 


**2.** ${u \_ n} \_{|\mathcal{P \_ n} }$ are uniformly $\alpha$-Hölder, that is there is $k\ge0$ such that $|u\_n(t) - u \_ n(s)|\le k|t-s|^\alpha$ holds for any $n\in\mathbb{N}$ and for any $t,s\in\mathcal{P} \_ n^M\, .$


Reason: if $\tilde u \_ n$ denotes the piece-wise interpolation of the nodes $\mathcal{P} \_ n^M$, then by concavity $\tilde u \_ n$ has modulus of continuity $k|t|^\alpha$ on $[a,b]$, and $\| u \_ n - \tilde u \_ n\| \_ \infty\le k\|\mathcal{P} \_ n\|^\alpha=o(1)$ as $n\to\infty$. Therefore $u$ has the same modulus of continuity $k|t|^\alpha$. 


