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

Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

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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: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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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: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

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: Let f be analytic in a region G. If f(g) lies on the parabola u=v^2, prove that f is constant in G.

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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: use the formal definition of limit to prove that thefunction is continuous at c.

A: Given: hx=2x-3 To Find: To prove that the given function is continuous at c=2

Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

Q: Let f:X → Y be function and g: X → fcX ×Y, g(x) = (x, f(x)). Prove that if g is a homeomorphism then…

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

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: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: Prove that L(f) ≤ 0 ≤ U(f). Given that the bounded function f : [a, b] → R has the property that…

A: Given that the bounded function f : [a, b] → R has the property that f(x) = 0 for all x ∈ [a, b] ∩…

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: Show that the function f defined by (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

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Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: Let f be differentiable on the closed interval [a, b] such that f′(a) = y1 and f′(b) = y2. If d lies…

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Q: Let f : [a, b] R be a continuous function on [a, b). Prove that if |f(x)| dx = 0 then f(r) = 0 for…

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Q: Let f:X Y be function and g:X→fcXXY, g(x) = (x, f(x)). Prove that if g is a homeomorphism then f is…

A: Given that, f:X→Y be function and g:X→f⊂X x Y defined byg(x) = (x,f(x))Given that, 'g' is…

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 g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

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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 f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: Let f:X Y be function and g:X →fcX XY, g(x) = (x, f (x)). Prove that if g is a homeomorphism then f…

A: Given :  g: X→f⊂X×Y is a homeomorphism.  So, by definition of homeomorphism, g is continuous also.…

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

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Q: Show that the function f defined by (x, y) = (1, – 1) f (x, y) : x2 + Y (x, y) + (1, – 1) x + y is…

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Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

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: Show that the function f defined by 1, (x, y) = (1, – 1) f (x, y) : .2 x + y (x, y) # (1, – 1) x + y…

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Q: Let p : X → Y and f : Y → X be continuous. Suppose that po f equals the identity map of Y. Prove…

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Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

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Q: If f (z) is analytic and f' (z) is continuous at all points inside and on a simple closed curve C,…

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Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f…

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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…

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Q: Find and describe the points in the space that the function f(x, y,z)=In(xyz) is continuous.

A: We have to find and describe the points in the space that the function f(x,y,z)=ln(xyz) is…

Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

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: Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not…

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Q: Suppose f : [a,b] → R is a continuous function. Prove that the direct image f([a,b]) is a closed and…

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Q: Show that the function f defined by 1. (x, y) = (1, –1) f (x, y) : x² + y .2 (x, y) + (1, –1) x + Y…

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Q: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

Q: Suppose that a continuous map g : U → W is tangent at a to a map f:U → W. Prove that f is also…

A: given that continuous map g:U→W is tangent at  a  to a map  f:U→W. we have to show that f is also…

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: Let f : R → R be a continuous function that is convex on (-0o, 0 and 0, 00) and has a local maximum…

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