RO-NMO 2009 Problem 124

Find all functions $f:[0,1]\longrightarrow [0,1]$ that are bijective, continuous and have the property that, for any continuous function $g:[0,1]\longrightarrow\mathbb{R} ,$ the following equality holds. $$\int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx$$