Prove that a space X is connected iff every continuous function F:X-->Y={0,1},Y is discrete is constant?

if f is continuous and Y is discrete, so f(X) is connected since f is continuous, and thus f(X) is not equal to Y so f is constant. For the converse use contra positive, so ume X is disconnected with separation A,B, and Y is discrete. Define a function f from X into Y as f(A)=1, f(B)=0, then f is continuous but not constant.