b) Let f :R R , then prove that the following three statements are equivalent. (i) f is continuous at 1, (ii) Given &>0 there exists &>0 such that (x, - 8.x, + 8)Cf(f(x)-E, f(x) + ɛ) (iii) For every sequence (x,) in R such that (x)→ x, we have (f(x))→ f(x,).

Q: - Let (fn)1 be a sequence of functions such that fnf on a set X. Which of the following condition(s…

A: X should be a closed interval.

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A: These statements refer to the second derivative of a function f(x) and describe how the function is…

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A:

Q: 3. Consider the function f(x) = sin x. (i) Is there a sequence of polynomials converging uniformly…

A:

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A: 1) is false and 2) is true.

Q: if f:[0,2] 3 R such that if x>0 f(x) = { ²2 if x<0 show that whether f is continuous in [0,2]

A:

Q: If f(x) is continuous everywhere, then If(x)| is continuous everywhere. TRUE O FALSE

A: If f(x) is continuous everywhere, then |f(x)| is continuous everywhere.True of false.

Q: Let f (x) = x + x3 + 2x 1. The function f is increasing on the following interval(s): (а, b) O [a,…

A:

Q: (A) True (B) False If f(x) is continuous, f(-1) = 2, and f(2)= -1, then f(x) has a root in [-1,2].

A: This question is related to intermediate value theorem .

Q: Given f(x) is continuous on its domain of (-00,-4) U(-4,00) and f" (x)= (x-8)7(x-1)² (x+4)3 A. x = 1…

A: Inflection points are those points where second derivative changes sign and those points at which…

Q: 5 - x, -1 sx 2 r - 1, 2 <xs 3 Discuss the continuity of g(x) = %3D

A:

Q: A function f is continuous on [-3, 5] and contains the points (-3, -4), (0, ƒ(0)), and (5, –3). If ƒ…

A:

Q: Define a function f: N → Z by n/2 (1-n)/2 f(n) = { (a) Show that f is one-to-one correspondence. (b)…

A: We need to show that if , then .Let's consider two cases:Case 1:  and  are both even…

Q: f(x) = x(x-1) for rational numbers          = 0 for irrational numbers  Prove f is continuous at x…

A: Given function is continuous at x= 0 and x=1. See the attachment.

Q: (4ах + 2b , X<1 2х — 3Ь,1 <х < 3 (2ах + 4b g(x) ,3 <x Find the values of "a" & “b" if gm is…

A: We will find the value of a and b as following-

Q: Q3) Test the continuity of the function f(x): -15x <0 0S X< 2 F(x) = 2х— 1 x+ 3 X = 2

A:

Q: Let f(x) = { 3, 1< x < 2 0, 2 < a <5 and Let F(x) = { 3x, 1<x < 2 6, 2 <x < 5 Show that wether the…

A:

Q: True or false: a function f: R → R is continuous at a point a ER if and only if there exists a…

A:

Q: Q1) Using the definition of continuity prove that the functions are continuous 1 fox)=x²³² +1 it x=2…

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Q: Let f(x)= 1 Use a 8- e proof to show that f is continuous at x = = 2. X-1

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Q: if x ≤ 0 if 0 2 Use the definition for g to evaluate the following. Show work where appropriate. 1,…

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Q: n a and b with f(c) = N. I thi

A: Mean value theorem: Rolles theorem If a real-valued function f is continuous on a proper  [a, b],…

Q: Consider the function f with domain (-2, : 3|22 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

A:

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: x 2) Discuss the continuity of the graph of f(x) given below on the closed interval [-4, 2] D(-4,6)…

A: This question is related to continuity of functions, we will solve it using given information.

Q: Prove that x = 0 is a discontinuity of each function (a) f(x) = 1 for x

A:

Q: b) Discuss the continuity conditions of the function- f(x) which shown in figure at x= 0,x = 1,x= 2…

A: