(related to Theorem: Mean Value Theorem For Riemann Integrals)
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
\[\int_a^bf(x)\cdot\phi(x)dx= f(\xi)\cdot \int_a^b\phi(x)dx.\]
