 Solution

Choose continuous boundary values equal to $\sin x$ on the lower side and zero on the other three sides. These values agree at all four corners, so they define a continuous function on the boundary of the rectangle. Any solution would satisfy the zero vertical-side conditions of part (a), whose Fourier sine series forces its lower boundary trace to be zero. This contradicts the prescribed value $\sin x$ for $0<x<\pi$. Hence these continuous Dirichlet data admit no solution in $C^0(\overline{R})\cap C^2(R)$.