(related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)

- Let $\phi\in T[a,b]$ be a step function.
- By definition of Riemann upper and Riemann lower integrals, we have $$\int_{a~*}^{b} \phi(x)dx = \int_a^b \phi(x)dx=\int_a^{b~*} \phi(x)dx.$$
- Thus, by definition, $\phi$ is Riemann integrable.∎

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983