5. Consider X with the discrete metric. (a) Show that every subset of X is open and closed. (b) Show that if f: X→Y is a function from X to another metric space Y, then f is continuous. (c) Which sequences converge in X?

5. Consider X with the discrete metric. (a) Show that every subset of X is open and closed. (b) Show that if f: X→Y is a function from X to another metric space Y, then f is continuous. (c) Which sequences converge in X?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5. Consider X with the discrete metric.
(a) Show that every subset of X is open and closed.
(b) Show that if f: X→Y is a function from X to another metric space Y, then f
is continuous.
(c) Which sequences converge in X?

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Consider X with the discrete metric. (a) Show that every subset of X is open and closed. (b) Show that if f: X→Y is a function from X to another metric space Y, then f is continuous.

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2. Use the fact ”if a function f is continuous on R and (xn) a sequence in A such that, xn → x, then f(xn) → f(x)”, to prove that function g defined by g(x) =(1 if x < 0, 2 if x = 0, 3 if x > 0. is not continuous at x = 0.
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