Problem

Source: Romania National Olympiad 2023

Tags: real analysis, integration, function, riemann integral



Let $a,b \in \mathbb{R}$ with $a < b,$ 2 real numbers. We say that $f: [a,b] \rightarrow \mathbb{R}$ has property $(P)$ if there is an integrable function on $[a,b]$ with property that \[ f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b]. \] Show that for all real number $t$ there exist a unique function $f:[a,b] \rightarrow \mathbb{R}$ with property $(P),$ such that $\int_{a}^{b} f(x) \text{dx} = t.$