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 10
 Problem Statement

(a) find all continuous functions ##f## satisfying ##\int_0^x f(x) \, dx = (f(x))^2+C## for some constant ##C≠0## assuming that ##f## has at most one 0.
(b) Also find a solution that is 0 on an interval ##(\infty,b]## with ##0 \lt b##, but nonzero for ##x \gt b##
 Relevant Equations
 ##\frac d {dx} \int_a^x {f(t)} \, dt = f(x)##
I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But that is the one value that ##C## cannot have, so I don't see how such a function can be a solution to the equation for ##x \lt b##...