Show the equivalence of the following three definitions of continuity. Let (X, d) (Y, d) be...

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Show the equivalence of the following three definitions of continuity. Let (X, d) (Y, d) be two metric spaces. A function f from X into Y is said to be continuous at a = X if: (1) For each & > 0, there exists a > 0 such that f(B(a; 5)) B(f(a); e) (2) For each neighborhood N of f(a) in Y there exists a neighborhood M of a in X such that f(M) CN (3) For each neighborhood N of f(a) in Y, f(N) is a neighborhood of a in X. Show the equivalence of the following three definitions of continuity. Let (X, d) (Y, d) be two metric spaces. A function f from X into Y is said to be continuous at a = X if: (1) For each & > 0, there exists a > 0 such that f(B(a; 5)) B(f(a); e) (2) For each neighborhood N of f(a) in Y there exists a neighborhood M of a in X such that f(M) CN (3) For each neighborhood N of f(a) in Y, f(N) is a neighborhood of a in X.

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