Find a function whose ordinate set satisfies a given property

by RoRi

January 31, 2016

Find a nonnegative function $f$ whose ordinate set on an interval has area proportional to the sum of the function values at the endpoints of the interval.

With reference to the previous two exercises here and here we write

$A(x) = \int_a^x f(t) \, dt = C(f(x) + f(a)) \quad \implies \quad \int_a^a f(t) \, dt = 2C f(a).$

But,

$\int_a^a f(t) \, dt = 0 \quad \implies \quad 2Cf(a) = 0 \quad \implies \quad C = 0 \text{ or } f(a) =0.$

If $C = 0$ then $\int_a^x f(t) \, dt = 0$ for all $x$ ; hence, $f(x) = 0$ . If $f(a) = 0$ , then $f(x) = 0$ for all $x$ since $a$ was arbitrary. Thus,