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Functional Equation with Integral and Derivatives

2550MasterCalculusFunctions

SHORTLISTED PROBLEMS FOR THE 2019 ROMANIAN NMO · 2019

Let $n$ be a positive integer. Find all functions $f: [0, \infty) \to \mathbb{R}$, which are $n$ times differentiable, their $n^{th}$ derivative is bounded below and $$ x \int_{x}^{x+1} f(t) \, dt = \int_{0}^{x} f(t) \, dt, $$ for every real $x \ge 0$.
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