On the Riemann integrability of the \(n\)-th local modulus of continuity (Q873796)

It is shown that if \(f: [a,b]\to \mathbb{R}\) is Riemann integrable, \(n\in\mathbb{N}\), \(\delta> 0\), then \[ \omega_n(\delta, f,x):= \sup\{|\Delta^n_h f(t)|: t,t+ nh\in[x-\delta, x+\delta]\cap [a,b], 0\leq h\leq 2\delta/n\} \] is also Riemann integrable over \([a,b]\) (\(\Delta^m_ hf=n\)th difference of \(f\)). If \(f\) is only bounded, then already \(\omega_n(\delta, f,x)_k\) and \(\sup\{|f(t)|: t\in [x-\delta, x+\delta]\cap [a,b]\}\) are Riemann integrable over \([a, b]\), where \(\omega_n(\dots)_k\) is obtained from \(\omega_n\) by replacing \(2\delta/n\) by \(2\delta k/n(k+ 1)\), \(k\in\mathbb{N}\). Also some characterizations of Riemann integrability of the type upper Riemann integral \(\overline\int^b_a \omega_1(\delta, f,t)\,dt\to 0\) as \(\delta\to 0\) are given.