Problem of the week

2020-19 Continuous functions

Let $$n$$ be a positive integer. Determine all continuous functions $$f: [0, 1] \to \mathbb{R}$$ such that
$f(x_1) + \dots + f(x_n) =1$
for all $$x_1, \dots, x_n \in [0, 1]$$ satisfying $$x_1 + \dots + x_n = 1$$.