4. Suppose X and Y are normed spaces, dim Y <0, f:X→Y is linear and f(x) = Y. If ker(f) is closed. Prove that f is continuous. G S

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: prove if x → y then xor y = y =

A: To prove : If x→y then x or y = y Lets use the help of truth table to prove the above statement.

Q: 73. Let z = f(x, y) .Show that g(x, y) fa (x, y)g(x, y) – f(x, y)g:(x, y) (g(x, y))² az and ax əz _…

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Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

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Q: 1. Show that the following functions f : R R are continuous. You should argue carefully from the…

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

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Q: 4. Prove that f(x, y, z) = 2 - 1 x² + y² is continuous in all its domain.

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Q: Suppose a 4D space exists for variables a, b, c, and d, where a is a function of b, c, and d, then,…

A: Given: A 4D space with variables a, b, c and d. Variable a is function of b, c and d. a=fb, c ,d

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Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: Let X~U (0,0), then P(X > X²) is

A: Given that X~U0,θ fx=1θ-0=1θ, 0≤X≤θ

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Q: Find the absolute maximum and minimum values of f on the set D. f(x, y) = x4 + y4 − 4xy + 6, D =…

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Q: Find the aven bounded by y=x and y =x+2

A: First draw the graph of the function and find the intersection point of the graph:

Q: Clarify that f f :(0,1) →R defined by f(x): not uniformly continuous. =-

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Q: A ve B iki küme, f : A → B bir fonksiyon ve X, Y C A olsun. a) f(X) – f(Y) C f(X – Y) olduğunu…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: Let (f (u, v,w) and (u = u(x, y, z), v = v(x, y, z), w = w(x, y, z) be differentiable functions)…

A: Given function f(u, v, w) and u = u(x, y ,z),  v = v(x, y, z) , w = w(x,y,z) are differentiable…

Q: 5. Prove that |xy| = |x|-|y| for x, y eR.

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Q: Let f (x, y) = Vy – 2x Evaluate f(3, 10)

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Q: Find the absolute maximum and minimum values of f on the set D where f(x, y) = x² + 4y² − 2x − 8y +…

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Q: 30 Chapter 4 Differentiation example of a function that fails to satisfy a Lipschitz condition at a…

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Q: Suppose f(x, y) and g(x, y) are continuously differentiable and dyf(x, y) = dxg(x, y). Suppose this…

A: Given: (a)Suppose f(x,y) and g(x,y) are continuously differentiable and…

Q: 1. Let f: [a,b] → R be a bounded function such that f(x) = 0 except at xo E (a,b).

A: A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is…

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: Let f(x, y, z) = yexyz², then fxz is: Select one: O (4xyz + 2x²y³z²)exyz² O None of these O (2xy² +…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnB = 0, B nC = ¢, µ(AnC) = 1,then µ(AUBUC) = 1.…

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Q: Find a norm of the function flt)= t+2t in space C^co13 %3D where normis given by formula ms x…

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Q: Q# 3(a) Let (X, Tx), (X, Ty), (X, 7z) be topological spaces, and let f: (X, 7x)(X, Ty) and g: (X,…

A: Given :-  Let X , τx , X , τY , X , τZ be topological spaces , and let f : X , τX → X , τY and  g :…

Q: EXAMPLE 2 The set {sint, cost} is linearly independent in C[0, 1], the space of all continuous…

A: The given set is sin t, cos t.We need to show that the set is linearly independent.

Q: 9. Find div F if F(x, y, z) = x²yi + xyzj - x²y²k. A. 2xy + yz - 2xy² B. x² + xy C. 2xy - xz D. 2xy…

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Q: Define f: Z x Z→ Z x Z by f((m, n)) = (3, 2n). Let A = {(n, 0): n ∈Z}. Fina f(A).

A: EXPLANATION : Given function  f:ℤ×ℤ→ℤ×ℤ defined by fm,n=(3,2n). given set A={(n,0):n∈ℤ}. putting the…

Q: Prove z = max (x,y) and z = min(x,y) are convex

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Q: Let k e Zt. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified…

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Q: Let f(x, y) = xe Ty", find fr, fy, fræ, fay and fyy-

A: