Integral functionals that are continuous with respect to the weak topology on W01,p(0,1)

For continuous (or, locally bounded Carathéodory) functions g:[0,1]×R→R we prove that the functional Φ(u)=∫01g(x,u(x))dx is weakly continuous on W01,p(0,1), 1⩽p<∞, if and only if g is linear in the second variable.