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up vote 0 down vote favorite Let $f:mathbb{R}rightarrowmathbb{R}$ be a measurable function. Show that if $f$ is continuous and fix $x_0in mathbb{R}$ , then $lim_{nrightarrowinfty}nint_{x_0}^{x_0+1/n}fdm=f(x_0)$ . Hint: Use the max-min theorem which says: For any continuous function $f$ on a compact set $[a,b]$ , there are points $x_{max},x_{min}in[a,b]$ such that $f(x_{max})geq f(x)geq f(x_{min})$ for all $xin[a,b]$ . real-analysis share | cite | improve this question edited yesterday KCd 16.5k 38 73 asked yesterday ...