- 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.

q.e.d

| | | | created: 2020-01-09 14:33:15 | modified: 2020-01-09 14:33:15 | by: *bookofproofs* | references: [581]

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983