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alexandræ · uniform continuity on \(f:E\subseteq\mathbb R\to\mathbb R\) functions differentiable almost everywhere
uniform continuity on \(f:E\subseteq\mathbb R\to\mathbb R\) functions differentiable almost everywheredecember 23rd, 2023
a week or two ago, i arbitrarily fixed myself the goal of characterising uniform continuity in a more calculatory way, in order to develop a better intuition about it, and help some students get the difference between this and simple continuity.
it started from an intuition that had to with oscillations that go faster and faster, either near a point or at ±∞. that's obviously partly wrong, as functions like \(\mathbb R.x\mapsto\mathbb R.x^2\) or \(\mathbb R^*.x\mapsto\mathbb R.\!(\frac1x)\) are fundamental counterexemples... or are they ? no because for \(\varepsilon>0,\) if we write \(\pi_\varepsilon:\mathbb R.x\to(\mathbb R/\varepsilon\mathbb Z).\!(x+\varepsilon\mathbb Z),\) then \(\pi_\varepsilon\) composed with any of these functions do exhibit these sorts of faster and faster oscillations, while stuff like \(\mathbb R.x\mapsto\mathbb R.x\) doesn't, and happens to indeed by uniformly continuous. while formalising this, it kinda had to do reciprocals, level sets, and uniform discreteness, but it led me to some very impractical stuff, so i took a little turn towards functions whose derivatives exist almost everywhere on their domain of definition.
in fact, i looked up stuff about mathematical oscillation on wikipedia at some point, mostly to find a sort of pointwise notion about it. i saw it had stuff to do with the modulus of continuity, which i completely forgot about but now remember from my functional analysis courses ; still though, that's not really pointwise either, so not what i was searching. so i came to the page that talked about mathematical oscillation, which we can apparently define as such :
$$\begin{array}{lllll}
\omega&:&E\subseteq\bigcup\limits_{A,\,B\,\subseteq\,\mathbb R}B^A&\longrightarrow&\bigcup\limits_{A\,\subseteq\,\overline{\mathbb R}}\mathbb R^A\\[10pt]
&&f:A\to B&\longmapsto&\begin{array}{lllll}\omega(f)&:&\overline A^{\,:\,\overline{\mathbb R}}&\longrightarrow&\overline{\mathbb R}\\&&z&\longmapsto&\limsup\limits_{\zeta\,\to\,z}f(\zeta)-\liminf\limits_{\xi\,\to\,z}f(\xi)\!\end{array}
\end{array}$$
the problem with that is that it'd be indeterminate if both the liminf and limsup were infinite with the same sign (that's the only reason why i had to add \(E\subseteq\) in its definition). thankfully, within the scope of the results i want to display, i just need to tweak it like so :
$$\begin{array}{lllll}
\varpi&:&\bigcup\limits_{A,\,B\,\subseteq\,\mathbb R}B^A&\longrightarrow&\bigcup\limits_{A\,\subseteq\,\overline{\mathbb R}}\mathbb R^A\\[10pt]
&&f:A\to B&\longmapsto&\begin{array}{lllll}\varpi(f)&:&\overline A^{\,:\,\overline{\mathbb R}}&\longrightarrow&\overline{\mathbb R}\\&&z&\longmapsto&\limsup\limits_{\zeta\,\to\,z}\left|f(\zeta)-\liminf\limits_{\xi\,\to\,z}f(\xi)\right|\!\end{array}
\end{array}$$
where \(-\infty+\mathbb R=\{-\infty\}\) and \(+\infty+\mathbb R=\{+\infty\}.\)
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alexandræ
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