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Sufficient condition for continuous and nondecreasing function to be constant

Source: Romanian District Olympiad 2011, Grade XII, Problem 3

October 8, 2018
functionIntegralRiemann sumreal analysis

Problem Statement

Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a continuous and nondecreasing function.
a) Show that the sequence (12n∑i=12nf(i2n))n≥1 \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} is nonincreasing.
b) Prove that, if there exists some natural index at which the sequence above is equal to ∫01f(x)dx, \int_0^1 f(x)dx, then f f is constant.
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