Proposition: Integrals on Adjacent Intervals

Let $a < b < c$, let $[a,c]$ be a closed real interval. The function $f:[a,c]\to\mathbb R$ is Riemann-integrable, if and only if the restrictions $f|_{[a,b]}$ and $f|_{[b,c]}$ are Riemann-integrable. In this case, we have

$$\int_a^c f(x)dx=\int_a^b f(x)dx + \int_b^c f(x)dx.$$

In particular, if we set $$\int_a^a f(x)dx:=0$$ and $$\int_a^b f(x)dx:=-\int_b^a f(x)dx,$$

then the above formula hold for all relative positions of the points $a,b,c$, if $f$ is Riemann-integrable in the interval $[\min(a,b,c),\max(a,b,c)].$

Proofs: 1

Definitions: 1

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