Conjecture Every continuous function $f(v_1,\dots,v_n)$ of $n$ arguments such that it lies in the closed interval $[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$, is symmetric for all permutations of arguments, and is homogeneous (with the degree $1$) necessarily is a generalized mean.
Help me to prove or disprove (with a counterexample) this conjecture. Is it a known result?