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Find a function whose ordinate set over an interval has area proportional to the length of the interval

by RoRi

Find a nonnegative function f whose ordinate set over any interval has area proportional to the length of the interval.

The area of the ordinate set of f over an interval [a,b] is given by \int_a^b f(t) \, dt. So, if it is proportional to the length of the interval then we have

    \[ A(x) = \int_a^x f(t) \,dt = C(x-a) \quad \implies \quad A'(x) = f(x) = C. \]

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