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: ) Prove that if f: RR is continuous at a, then so is f.

A: Continuous function:                                      A function f from R to R is…

Q: Suppose f: (0, 1)→ (0, 1) is a surjective monotonic function. True or false: The function f is…

A: f: 0,1→0,1 is a surjective monotonic function.

Q: The coefficient of (z – T)² in the Taylor series expansion of sin z if z+T f(z)={z-T -1 if z=T…

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Q: Definition A function f(x) is continuous from the right at a number a if... And f(x) is continuous…

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Q: Let f(z) be an entire function such that |f(z)| < 1 on the unit circle and f(a) = 0 for z - a some a…

A: Here given f(z)≤1 and we have to prove f(z)≤z-α1-α¯z Where z lies on unit circle and f(α)=0 for some…

Q: Let A, B be sets and f : A → B a function. Let A' CA and B' C B. (a) Prove that f¯¹[ƒ[A']] 2 A'.…

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Q: Use the Bolzano Weierstrass Theorem to prove that a continuous function ƒ : [a, b] → R attains its…

A: The Bolzano-Weierstrass Theorem states that every bounded sequence of real numbers has a convergent…

Q: 5. Let f: Z → Z be defined by f(n) = n² + 1. (a). Is f a one-to-one function? Explain. (b). Is f an…

A: Given- The function is f(n)=n2+1 and f:ℤ→ℤ. Definition- Let a function from X to Y, One-to-one: If…

Q: (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f:…

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Q: If f : X → Y and g: Y→ Z are functions such that g of is one-to-one, then f is one-to-one.

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Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: Function f is defined by the equation above for all pairs (x,y) of real numbers such that f(x,y) is…

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Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

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

A: See the attachment.

Q: Let f : R → R be a function given by f(x) = 2x – 1 (a) Determine whether f is bijective or not. (b)…

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Q: Question 1(a). Give at least three examples of continuous functions classes with justification and…

A: (a) Take f(x)=x2 in the interval [0,∞) The given function is continuous but not uniformly…

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: Theorem 7.10 (Pasting lemma). Let X = A U B, where A, B are closed in X. Let f : A → Y and g : B → Y…

A: Let X=A∪B, where A, B are closed in X. Let f: A→Y and g: B→Y be continuous functions that agree on…

Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: Let ƒ : D → R be a function and let xo be a limit point of D that also belongs to D. Show that the…

A:

Q: Determine whether the function f : R → R defined by f(x) = e" is surjective. If not, determine its…

A: A function is said to be surjective/onto when range is equal to codomain of function. Given function…

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

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Q: Let f:X Y is a function and let (B: it e A) be an indexed family of Subset of Y. Show that a)(Ua B)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider the function g : ZxZ → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

A: (a) we see (5, 0) and (0, 9) are different elements in ZxZ.      But, g(5, 0) = (4.5.0, 9.5+5.0) =…

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…

Q: 1.) Which of the following are TRUE about differentiable function: I If rx) exists, then the…

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Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

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Q: Suppose f and g are differentiable functions on an interval I. Show that if f′(x) = g′(x) for all x…

A: It is given that, f′(x) = g′(x) for all x ∈ I and f(x0) = g(x0) for some x0 ∈ I.

Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

Q: A function f: Z× Z → Z is defined as f(m,n) = 2n-4m. Verify whether this function is injective and…

A: Here we have given a function f , we have to check, is this function injective and surjective?

Q: If f,g continuous on [a,b] and f(a) g(b). Show that there exists c ∈ (a,b) such that f(c) = g(c)

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Q: Consider the function h : Z x Z → Z ×Z defined by h(x, у) — (2ху, 7x — Зу). (a) Is h injective? (b)…

A: