An integral mean value theorem for regulated functions (Q1374573)

The main result is the following: Let \(f,g\) be functions on \([a,b]\), let \(f\) be non-negative and integrable and let \(g\) have right and left limits at each point. Then there is a \(\theta \in (a,b)\) and \(g^*(\theta)\) such that \(\min [g(\theta_+),g(\theta_-)]\leq g^*(\theta)\leq \max [g(\theta_+),g(\theta_-)]\) and \(g^*(\theta)\int_a^b f(t) dt=\int_a^b g(t)f(t) dt\).