A property for real valued functions

Question

Let $f$ be a real valued continuous function over a compact topological space $X$, say [0, 1], I am looking for a condition on $f$ such that if there exist real valued continuous functions $g_1,...,g_n , f_1,...,f_n$ over $X$ with $f=f_1g_1+...+f_ng_n$, then there exist $i , j\in\{ 1,...n \} $ and a real valued continuous functions $h$ and $k$ over $X$ such that $f=hf_i$ and $f=kg_j$?(that is, if a function is a linear combenition of some functions with functional-coefficients then that function is a multiple of one of them with functional-coefficient)