Question 3 (3.1) Let X = {a, b, c} and let 7 = {0, {a} , {b} , {a,b}} be a topology on X. Let f : X → X be the function defined by f(a) = a, f(b) = e, and f(c) = b. Is f continuous at : (a) a? (b) Ь? (c) c?

Q: Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3),…

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Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

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Q: Prove Theorem 7.3.1(b): If fis any function from O a set X to a set Y, then Iy ° f = f, where Iy is…

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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: 5. Consider the function f (X, J) → (Y, U), where (X, T) and (Y, U) are two topological spaces.…

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Q: For a function f: A → B and subsets C and D of A, prove that f(CUD) = f(C) Uƒ(D).

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Q: Let f: XY and g: Y→ V be functions. Show the following: (a) f is injective 3h: Y→ X such that hof=…

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Q: A) Let (R, T) be the usual topological space and f: R → R s.t. f(x) = 2x + 2.Show that f is…

A: Given that (ℝ,τ) is the usual topological space (the standard euclidean topology over ℝ). Also given…

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: Let A = R - {1} and define f: A→A by f(x) = (a) Find fo f(x). (b) Find fofo f(x). (c) Find f-¹(x).…

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Q: Let f : [a, b] –→ R be twice differentiable and let f(a) = f(b) = 0 and f(c) > 0, where cE (a, b).…

A: Step:-1 Mean Value Theorem:-  Let a function f : a, b →ℝ be such that 1. f is continuous on a, b and…

Q: Exercise 1. Let f : A → B be a function with X a subset of A. Prove that if f(X) = 0, then X = 0.

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: Theorem (6) If f is bounded and R – I on [a, b], f(x) > 0, Vx € [a, b] then ſ f 2 0

A: Prove that: If f is bounded and R-I on [a, b], fx≥0 for all x∈a, b , then ∫f≥0,

Q: Let E = {0, 2, 4, 6, ...}, define ƒ: E × E → Z¹ by the formula f(m, n) = 4m5¹, \ (m, n) € Ɛ × Ɛ.…

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Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

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A: In the given question given that ζa be a particular point topology on X={a,b,c,d}.so first of all we…

Q: Let f: X→ Y and g: Y→ V be functions. Show the following: (a) fis injective 3h: Y→ X such that ho f…

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Q: Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

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Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: 1. Show that the following maping f are linear: f : R² → R² defined by f(x, y) = (x+ y,x).

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Q: Let N C C be a domain (that is, a connected open subset), and let f: N → C be holomorphic. Prove…

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Q: (a) Assume that f: A onto B and g: B → A. Prove that g = f-¹ iff g of = IA or fog = IB (Theorem…

A: As per the policy we are solving question (a) for you .For the solution of (b) please repost the…

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Q: Consider the following functions. (c) f: GL₂(R) → RX, f + ((ad)) (($)) (d) f: GL₂(R) → R, ƒ (e) f:…

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Q: Let g:BX C- C(with BØ) be the function given by g (x, y) = y. (a) Prove that g is surjective. (b)…

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Q: Define g: (-0, ∞) → [-1,1] by f(x) = sin x ,Vx E R. This function is onto. %3D

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Q: Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

A: Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Q: 4. Let C₁ be the set of all continuous real-valued functions on I = [0, 1]. (a) Show that d : C₂ ×…

A: According to our guidelines I can answer only Question no. 4(a).

Q: - Let A = {1, 2, 3, .., 103. Consider f:P(A) → N given by f(B) = |B|. That is, f takes B subset of A…

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Q: Let h : C N x N where C CZ given by h(m) = (m2 +1, m2 + 2). Choose all subsets C' such that h is an…

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