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Proving Riemann-integrability

Source: Romanian District Olympiad 2014, Grade 12, P1

June 15, 2014
functionfloor functionlimitintegrationreal analysisreal analysis unsolved

Problem Statement

For each positive integer nn we consider the function fn:[0,n]→Rf_{n}:[0,n]\rightarrow{\mathbb{R}} defined by fn(x)=arctan⁡(⌊x⌋)f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} , where ⌊x⌋\left\lfloor x\right\rfloor denotes the floor of the real number xx. Prove that fnf_{n} is a Riemann Integrable function and find lim⁡n→∞1n∫0nfn(x)dx.\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.
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