A function f: S -> Y is continuous on a subset A of S if and only if for every ε > 0, there exists a δ > 0 such that d_Y(f(x),f(y)) < ε for all x,y in A with d_S(x,y) < δ.
A function f: S -> Y is continuous on a subset A of S if and only if for every ε > 0, there exists a δ > 0 such that d_Y(f(x),f(y)) < ε for all x,y in A with d_S(x,y) < δ.