  Let X be a topological space, and let C(X) be the complex of singular
cochains on X with real coefficients. We denote by Cc(X) the subcomplex given
by continuous cochains, i.e. by such cochains whose restriction to the space of
simplices (endowed with the compact-open topology) defines a continuous real
function. We prove that at least for "reasonable" spaces the inclusion of Cc(X)
in C(X) induces an isomorphism in cohomology, thus answering a question posed
by Mostow. We also prove that such isomorphism is isometric with respect to the
L^infty-norm on cohomology defined by Gromov. As an application, we discuss a
cohomological proof of Gromov's proportionality principle for the simplicial
volume of Riemannian manifolds.
