A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0 +r) = {xo}. That is, the only point of D in (xo – r, xo +r) is xo itself (a) Prove that a point xo in D is either an isolated point or a limit point of D. (b) Suppose that xo is an isolated point of D. Prove that every function f : D → R is continuous at xo. (c) Let D = { |n E N} and define f : D → R by f(x) = !. Is f continuous at every x in D?

A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0 +r) = {xo}. That is, the only point of D in (xo – r, xo +r) is xo itself (a) Prove that a point xo in D is either an isolated point or a limit point of D. (b) Suppose that xo is an isolated point of D. Prove that every function f : D → R is continuous at xo. (c) Let D = { |n E N} and define f : D → R by f(x) = !. Is f continuous at every x in D?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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