The present note proves Riemann integrability of the $n$-th local modulus of continuity which is used within the definition of averaged moduli of smoothness ($\tau$-moduli). In addition it is shown that an averaged supremum norm ($\delta$-norm) can be calculated using the Riemann integral.
"On the Riemann integrability of the $n$-th local modulus of continuity." Funct. Approx. Comment. Math. 34 7 - 17, 2005. https://doi.org/10.7169/facm/1538186583