KöMaL 2024 A883

Let $J \subsetneq I \subseteq \mathbb{R}$ be non-empty open intervals, and let $f_1$, $f_2$, $\ldots$ be real polynomials satisfying the following conditions:

• $f_i(x) \geq 0$ for all $i \geq 1$ and $x \in I$,
• $\sum_{i=1}^\infty f_i(x)$ is finite for all $x \in I$,
• $\sum_{i=1}^\infty f_i(x)=1$ for all $x \in J$. Do these conditions imply that $\sum_{i=1}^\infty f_i(x)=1$ also for all $x \in I$?