Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not equal to y }. a. Prove that d is a metric on R. b. Prove that for any a in R, {a} is an open set under the metric d. c. Let X = R with the usual Euclidean distance and Y = R with the metric d defined above. Prove that f maps X to Y defined by f(x) = x is not continuous.

Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not equal to y }. a. Prove that d is a metric on R. b. Prove that for any a in R, {a} is an open set under the metric d. c. Let X = R with the usual Euclidean distance and Y = R with the metric d defined above. Prove that f maps X to Y defined by f(x) = x is not continuous.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter8: Areas Of Polygons And Circles

Section8.4: Cicumference And Area Of A Cicle

Problem 22E: What is the limit of mRTS if T lies in the interior of the shaded region?

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Suppose that we define a function d that maps R² to R by

d(x,y) = { 0,. if x = y

1, if x is not equal to y }.

a. Prove that d is a metric on R.

b. Prove that for any a in R, {a} is an open set under the metric d.

c. Let X = R with the usual Euclidean distance and Y = R with the metric d defined above. Prove that f maps X to Y defined by f(x) = x is not continuous.

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