An integral mean value theorem for regulated functions

Abstract

An elementary but useful mean value theorem for integrals asserts that if \(f\) is a non-negative integrable function on \([a,b]\) and \(g\) is continuous there, then there is a \(\theta\in(a,b)\) such that \[g(\theta)\int^b_af(t)dt=\int^b_ag(t)f(t)dt.\] It does not seem to have been observed that this result has an equally useful extension to regulated \(g\), i.e, \(g\) which have right and left limits at each point.