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Cute characterization for Riemann-integrable functions

Source: Science ON 2021 grade XII/3

March 16, 2021
integrabilityriemann integrabilityfunctionreal analysis

Problem Statement

Define E⊆{f:[0,1]→R∣f is Riemann-integrable}E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\} such that EE posseses the following properties:\\ <spanclass=′latex−bold′>(i)</span><span class='latex-bold'>(i)</span> If ∫01f(x)g(x)dx=0\int_0^1 f(x)g(x) dx = 0 for f∈Ef\in E with ∫01f2(t)dt≠0\int_0^1f^2(t)dt \neq 0, then g∈Eg\in E; \\ <spanclass=′latex−bold′>(ii)</span><span class='latex-bold'>(ii)</span> There exists h∈Eh\in E with ∫01h2(t)dt≠0\int_0^1 h^2(t)dt\neq 0.\\ Prove that E={f:[0,1]→R∣f is Riemann-integrable}E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}. \\ (Andrei Bâra)
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