1. Consider a set X with discrete metric and Y an arbitrary metric space. Consider a...

 

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1. Consider a set X with discrete metric and Y an arbitrary metric space. Consider a function f X→ Y Which functions f will be continuous? Justify your conclusion. 2. Suppose X, Y and Z are metric spaces; f : X→ Y and g: Y→ Z are continuous functions. Prove that go f: X→ Z is continuous. 3. Let X and Y be two metric spaces and f: X Y be uniformly continuous. Prove that for any Cauchy sequence {n: n = w} in X, the sequence {f(n): new} is a Cauchy sequence in Y. 1. Consider a set X with discrete metric and Y an arbitrary metric space. Consider a function f X→ Y Which functions f will be continuous? Justify your conclusion. 2. Suppose X, Y and Z are metric spaces; f : X→ Y and g: Y→ Z are continuous functions. Prove that go f: X→ Z is continuous. 3. Let X and Y be two metric spaces and f: X Y be uniformly continuous. Prove that for any Cauchy sequence {n: n = w} in X, the sequence {f(n): new} is a Cauchy sequence in Y.

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