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Proposition: Linearity and Monotony of the Riemann Integral for Step Functions

Let \(\phi,\psi\in T[a,b]\) be step functions. The Riemann integral for step functions fulfills the following rules:

Linearity Rules:

\[\int_a^b(\phi+\psi)(x)dx=\int_a^b\phi(x)dx+\int_a^b\psi(x)dx\] \[\int_a^b(\lambda\cdot \phi)(x)dx=\lambda\cdot\int_a^b\phi(x)dx\quad\quad(\text{for all }\lambda\in\mathbb R)\]

Monotony Rule:

\[\phi\le \psi\Rightarrow \int_a^b\phi(x)dx\le \int_a^b\psi(x)dx\]

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2

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References

Bibliography

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