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integral of f is equal to two integrals of f

Source: Romanian District Olympiad 2004, Grade XII, Problem 4

October 7, 2018
Integralreal analysiscalculusintegration

Problem Statement

Let a,b∈(0,1) a,b\in (0,1) and a continuous function f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} with the property that \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt, \forall x\in [0,1] .
a) Show that if a+b<1, a+b<1, then f=0. f=0. b) Show that if a+b=1, a+b=1, then f f is constant.
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