Suppose that A is a subspace of a topological space X, Y is a Hausdorff space, and f:A->Y is a continuous function. Suppose that g1:Cl(A)->Y and g2:Cl(A)->Y are continuous functions such that g1|A=g2|A=f, where Cl(A) is the closure of A. Prove that g1=g2. Solution To prove that g1 = g2 g1 and g2 are given to be continuous and g1/A = g2/A =f i.e. g1 and g2 are equal around the closure of A. Also g1 and g2 are defined for domains closure of A Hence g1=g2 for every point in its domain Hence it follows g1=g2.