3. Let X and Y be metric spaces. Let f :X → Y be a function. Definition: If f is discontinuous at a, we say that f has a removable discontinuity at a provided that there exists a function g : X → Y such that g is continuous at a and 9(1) = f(x) for all I#a. (a) Give an example of a function f: R → R with a removable discon- tinuity.

Q: Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are…

A: If the product AB is defined, the entry in row i and column j of AB is the sum of the products of…

Q: R.2. Let X be the collection of all functions f :R→R. Show that there exists a one-to-one function G…

A: It is given that X is the collection of all functions f: ℝ→ℝ. Show that there exists one-to-one…

Q: 2. Let (X,r) and (Y.A) be two topological spaces. Let B be a base for A. Prove that a function f…

A: Given below the detailed solution

Q: 1.3.2. Let f: X → Y be a function, let B be a subset of Y, and let {B;}ie1 be a family of subsets of…

A: Let f:X→Y be a function, let B be a subset of Y and Bii∈I be a family of subsets of Y.  To prove…

Q: (2.9.13, Given continuous functions f, g: X F on a metric space X, prove the following statements.…

A: Note: (a) We say that f and g are continuous if and only if df(x),f(c)<ε⇔dx,c<δ The aim is to…

Q: 6 language of a function f being continous at a point zo € dom(ƒ). Let f(x) = 2x² +1. Using the e- &…

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Q: Suppose f X→ Y, and A, B C X. Prove that (a) f(AUB) = ƒ(A) U ƒ(B) (b) ƒ(ANB) ≤ ƒ(A) Ñ ƒ(B)

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Q: Let f: (a) Is f one-to-one? Prove your answer. (b) Is fonto? Prove your answer. (c) Is there a…

A:

Q: Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) 3D х is…

A: Suppose U is an open set in X. Its inverse image is then i−1(U) = {x ∈ A : i(x) ∈ U}          = {x…

Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: Let I  be an interval and let f : I → R be a continuous, injective function. Show that f is…

A:

Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

A:

Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: continuous map со

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Q: Prove: There exist functions f, g, h in F such that g o f = h o f, but g is not equal to h.

A: The objective is to prove there exist functions f, g, h in F such that g o f = h o f, but g is not…

Q: Prove, using the definition of continuity from the topological point of view, that the constant…

A:

Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

A: Injective ( one to one): If x1 not equal to x2 then fx1 not equal to fx2. If x1 = x2 then fx1 = fx2.…

Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) =…

A: Given, X, Y be topological spaces and y0∈Y. The constant map f:X→Y defined by fx=y0 Shows in the…