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Continuous increasing function integral equation

2050SpecialistCalculusFunctions

66th ROMANIAN MATHEMATICAL OLYMPIAD

Determine the continuous increasing functions $f: [0, \infty) \to \mathbb{R}$ satisfying $$\int_{0}^{x+y} f(t) dt = \int_{0}^{x} f(t) dt + \int_{0}^{y} f(t) dt,$$ for all non-negative real numbers $x$ and $y$.
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