a) Show that, any function f: X → Y is continuous if X have discrete topology.

Q: Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

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Q: Prove that f(x) = |x| is continuous at all points c in ℝ.

A: Continuity of a function can be done by many methods as,

Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-Df(x)+(y) vx, yER. Show that…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

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Q: If a subadditive functional p on a normed space X is continuous at 0 and p(0) = 0, show that p is…

A: Given: A sub additive functional p on a normed space X is continuous at 0 and p0=0. To prove: p is…

Q: Show that a uniformly continuous function on (a, b) is bounded.

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Q: If f is continuous on a closed interval [a, b], then f attains an absolute maximum value and an…

A: According to questionf is continuous on a closed interval [a,b] then f attains an absolute…

Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: Let A = {0, 1} with the discrete topology and let f : A → R be defined by f(0) = −1, f(1) = 1. Show…

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Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

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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 f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

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Q: prove that any Continuouse function [a,b] en is measurable.

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Q: Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff then f is a…

A: To prove , Let f: X→ Y be a bijective continuous function. If X is compact and Y is Hausdorff then f…

Q: With X a topological space, is the function defined as A:X →X × X,x → (x, x) continuous? Investigate

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Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -> R is a Lipschitz continuous function defined on X . To show : f is…

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

Q: Let X ų be topological spaces and f: X→Y be a continuous map. If X is completely normal then Y is…

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Q: Assume that g : R → R is a periodic continuous function. Prove that f(x) = 2x + g(x) is uniformly…

A: The complete solutions are given below

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

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

Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: Let f be a continuous functio to R. Let a and b be two distinct ntained in U and fbe differentiable…

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Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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Q: Show using Definition 4.3.1(Continuity) that if c is an isolated point ofA ⊆ R, then f : A → R is…

A: If c∈A⊆ R is an isolated point of A ⊆ R. To prove  f : A → R is continuous at c. Let ∈>0 , Since…

Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

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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 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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Q: State whether each of the following statements is true or false, in either case…

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Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

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Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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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: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

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Q: Consider the continuous function h with h'(x). In(x) +6 3 3.1 Find and represent the domain of the…

A: We have to find domain

Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

A: Continuous Function: Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y is continuous…

Q: Exercise 6. Prove that if f [a, b] : interval. → R is continuous, then f([a, b]) is a closed and…

A: 6. Given that f : a,b→ℝ is a continuous function. To find: f a,b is closed and bounded interval.…

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: Let f: [a, b] → [a, b] be continuous. A fixed point of f is a member x E [a, b] such that x = f(x).…

A: Here we use IVP theorem to prove this

Q: Q3: A) If f is continuous function on a compact metric space X Show that £ieunifone

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Q: Give an example of a continuous function T, and an open set O in a metric space (X, d) such that…

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

Q: Show that the space C(R) of all continuous functions defined on the real line is an…

A: According to the given information, it is required to show that C(R) of all continuous functions…

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: f (x) = %3D x² +1

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.