(2) Define a function f : R → R by f(x) = X if x = for some n = N; otherwise. (a) Using the d-e characterization of continuity, prove that f is continuous at x = 0. (b) For which a ER is f continuous at x = a? Prove your answer.

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A: When a function has a removable discontinuity at a point, it means that the function is not defined…

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A: f(x) = x Domain of f(x), x ≥ 0x ∈[0, ∞) At x = 0, f(0) = 0f(0+h) = lim h→0 0 + h = 0f(0) = f(0+h)…

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A: Given,         The function f is continuous on 0,4. f0=-2 and f4= 5.         There exist a number c…

Q: Let g(x) = 3√x. (a) Prove that g is continuous at c = 0. (b) Prove that g is continuous at a point c…

A: a) limx→0g(x) = limx→0 x1/3                     =0Also , g(0) = 0Hence g(x) is continous at x=0 or c…

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A: The function f(z) = (2z + 1)(z2 - 1)  at the point z = - 1+ 2i is  ? Continious and differentiable.…

Q: (b) Consider the function f : R → R, f(x) = { (i) Prove that f is discontinuous at 0. (ii) Prove…

A:

Q: 3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1].…

A:

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A: Calculate the limit:

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Q: (x+1 Q. #7 For the function g(x) = 3 if x 2 constant "b" that makes g(x) continuous at x=2 Final…

A: g(x) is continuous    here we find value of b by using condition of continuous function

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A: We take an example g(x, y)=(x+y)2x2+y2 . It is continuous w.r.to x  and y but jointly not

Q: Q2: Solve the following differential equations. dx 2х — Зу X(0) = 8 dt dy = y —D у — 2х Y(0) = 3 dt

A: Image is attached.

Q: Use the definition of continuity to determine whether the shown function is continuous at a = 0.

A:

Q: Q1: A) Let f: R R be a function such that f(x) D sign(x) Discuss the %3D continuity of f on R.

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A: Given Data: Show that: It is possible for a passive ADP agent to learn a transition model for which…