MathDB

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)\ge 0

for all

i\ge 1

and

x\in I

, [*]

\sum\limits_{i=1}^\infty f_i(x)

is finite for all

x\in I

, [*]

\sum\limits_{i=1}^\infty f_i(x)=1

for all

x\in J

.
Do these conditions imply that

\sum\limits_{i=1}^\infty f_i(x)=1

also for all

x\in I

?
Proposed by András Imolay, Budapest

Problem Statement