# Proof

Let $$m$$ and $$M$$ be the values of the infimum and the supremum of all values, which the function $$f$$ takes on the closed real interval $$[a,b]$$, formally: $\begin{array}{rcll} m&:=&\inf&\{f(x):x\in[a,b]\}\\ M&:=&\sup&\{f(x):x\in[a,b]\} \end{array}$

By hypothesis, $$\phi(x)\ge 0$$ for all $$x\in[a,b]$$. Thus, we get $m\cdot \phi(x)\le f(x)\cdot\phi(x)\le M\cdot\phi(x)\quad\quad\text{for all }x\in[a,b].$ It follows from the linearity and monotony of the Riemann integral. $m\cdot \int_a^b\phi(x)dx\le \int_a^bf(x)\cdot\phi(x)dx\le M\cdot \int_a^b\phi(x)dx.$

Therefore, there exists $$\mu\in[m,M]$$ with

$\int_a^bf(x)\cdot\phi(x)dx= \mu\cdot \int_a^b\phi(x)dx.$

By hypothesis, $$f:[a,b]\mapsto\mathbb R$$ is a continuous functions. By virtue of the intermediate value theorem, $$f$$ takes any value between $$f(a)$$ and $$f(b)$$, in particular the value $$\mu$$. Thus, there exist $$\xi\in[a,b]$$ with $$f(\xi)=\mu$$ and

$\int_a^bf(x)\cdot\phi(x)dx= f(\xi)\cdot \int_a^b\phi(x)dx.$

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### References

#### Bibliography

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