If f A B and g: C B such that A C C and for...

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If f A → B and g: C B such that A C C and for each a € A we have f(a) = g(a) then g extends f. We also say that f extends to g. Assume that f: S→ R is a uniformly continuous function defined on a subset S of a metric space M. (a) Prove that f extends to a uniformly continuous function 7:5 → R. (b) Prove that is the unique continuous extension of f to a function defined on 5. (c) Prove the same things when R is replaced with a complete metric space N. If f A → B and g: C B such that A C C and for each a € A we have f(a) = g(a) then g extends f. We also say that f extends to g. Assume that f: S→ R is a uniformly continuous function defined on a subset S of a metric space M. (a) Prove that f extends to a uniformly continuous function 7:5 → R. (b) Prove that is the unique continuous extension of f to a function defined on 5. (c) Prove the same things when R is replaced with a complete metric space N.

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