The following proposition shows that the Riemann integrable functions on an arbitrary closed interval $[a,b]$ for a vector space and that the integral is monotonic.

Proposition: Linearity and Monotony of the Riemann Integral

Let \(f,g:[a,b]\mapsto\mathbb R\) be Riemann integrable functions on the closed interval \([a,b]\) and let $\lambda\in\mathbb R$. Then the functions $f+g$ and $\lambda f$ are Riemann-integrable and their Riemann integral fulfills the following rules:

Linearity Rules:

\[\int_a^b(f+g)(x)dx=\int_a^bf(x)dx+\int_a^bg(x)dx\] \[\int_a^b(\lambda\cdot f)(x)dx=\lambda\cdot\int_a^bf(x)dx\quad\quad(\text{for all }\lambda\in\mathbb R)\]

Monotony Rule:

\[f\le g\Rightarrow \int_a^bf (x)dx\le \int_a^bg(x)dx\]

Proofs: 1

Proofs: 1 2
Propositions: 3

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