Question Number 30412 by abdo imad last updated on 22/Feb/18
![f is a function increazing(or decreazing)on ]0,1] prove that lim_(n→∞) (1/n)Σ_(q=1) ^n f((q/n))=∫_0 ^1 f(t)dt.](https://www.tinkutara.com/question/Q30412.png)
$$\left.{f}\left.\:{is}\:{a}\:{function}\:{increazing}\left({or}\:{decreazing}\right){on}\:\right]\mathrm{0},\mathrm{1}\right] \\ $$$${prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}}{{n}}\sum_{{q}=\mathrm{1}} ^{{n}} {f}\left(\frac{{q}}{{n}}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({t}\right){dt}. \\ $$$$ \\ $$