Question 155
Let $f : [0, 1] \to \mathbb{R}$ be an integrable function with $f(0) = 0$, such that
$$
\int_{0}^{1} \left( f\left(\frac{x}{n}\right) + f\left(\frac{x}{n+1}\right) \right) dx = 0, \quad \text{for any } n \in \mathbb{N}^{*}.
$$
a) Give an example of a function with the above property, which is primitivable and is non-identically vanishing on any interval included in $[0, 1]$.
b) If $f$ is continuous on $(0, 1]$, show that $f$ is primitivable on $[0, 1]$.
$$
\int_{0}^{1} \left( f\left(\frac{x}{n}\right) + f\left(\frac{x}{n+1}\right) \right) dx = 0, \quad \text{for any } n \in \mathbb{N}^{*}.
$$
a) Give an example of a function with the above property, which is primitivable and is non-identically vanishing on any interval included in $[0, 1]$.
b) If $f$ is continuous on $(0, 1]$, show that $f$ is primitivable on $[0, 1]$.