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f_(n+1)(x)=int^x_0 f_n(t)dt and f_n(1)=0 for all n

Source: VJIMC 2001 2.2

July 21, 2021
calculusintegration

Problem Statement

Let f:[0,1]→Rf:[0,1]\to\mathbb R be a continuous function. Define a sequence of functions fn:[0,1]→Rf_n:[0,1]\to\mathbb R in the following way: f0(x)=f(x),fn+1(x)=∫0xfn(t)dt,n=0,1,2,….f_0(x)=f(x),\qquad f_{n+1}(x)=\int^x_0f_n(t)\text dt,\qquad n=0,1,2,\ldots.Prove that if fn(1)=0f_n(1)=0 for all nn, then f(x)≡0f(x)\equiv0.
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