MathDB

Determine all functions

f : \mathbb{R} \to \mathbb{R}

that satisfy the following two properties. (i) The Riemann integral

\int_a^b f(t) \mathrm dt

exists for all real numbers

a < b

. (ii) For every real number

x

and every integer

n \ge 1

we have

f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt.

B3