Consider a variable X with a continuous uniform distribution in the interval [a,b] and a variable Y that is fully dependent on X, i.e., p(Y=y | X=x)=δ(x=y), where δ is a delta distribution with peak x. What is the entropy H(X,Y) of the joint distribution?

Intuitively, samples from the joint distribution should have a uniform distribution along the diagonal in [a,b]^2, so the entropy should be finite, but I can't really figure it out.