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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Q: Let. f be q into medric Captinyous. marping of a medric rpare %.

A: Matric space

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A: Given set A={1,2}

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Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

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A: Let us prove above result

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Q: Show that it A is subset of X any with the discre te metric then %3D

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A: Linear Algebra

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A: Metric space function

Q: Consider the following mappings on RXR: d1(r, y) = e , d2 = |2.r – 2y|l. and d3 = v2r - yl. None of…

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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.