Question 67
Let $a, b \in \mathbb{R}$, with $a \lt b$, be two arbitrary real numbers. We say that a function $f : [a, b] \to \mathbb{R}$ has property ($\mathcal{P}$) if it is an integrable function on $[a, b]$, such that
$$
f(x) - f\left(\frac{x+a}{2}\right) = f\left(\frac{x+b}{2}\right) - f(x) \quad \text{for any } x \in [a, b].
$$
Show that for any real number $t$ there is a unique function $f : [a, b] \to \mathbb{R}$ with the property ($\mathcal{P}$), such that $\int_a^b f(x) dx = t$.
$$
f(x) - f\left(\frac{x+a}{2}\right) = f\left(\frac{x+b}{2}\right) - f(x) \quad \text{for any } x \in [a, b].
$$
Show that for any real number $t$ there is a unique function $f : [a, b] \to \mathbb{R}$ with the property ($\mathcal{P}$), such that $\int_a^b f(x) dx = t$.