) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and justify your answer) such that is continuous at x = 0. exists at x = 0.

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A: Topic- Limits

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Q: Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

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Q: Prove that if f, g :R → R are continuous at x E R, then ƒ+gis continuous at x E R as well.

A: By using the definition of continuous function this above statement can be easily proved.

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Q: : Iff (z) is analytic at zo. (a) prove that it must be continuous at zo.

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Q: For real number p, let f and g be defined by f(x) =e¯*x³ and g(x) = 1/xP, respectively, on [1,0).…

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Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

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Q: Assume f(x) is continuous on [a, b]. Prove that if f(x) > 0 for all x € [a, b], then S. f(x) dx > 0.…

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Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [0,…

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Q: 3. Consider the function g(x) defined by | 22 sin(1/x) if x > 0, g(x) = –x² if x < 0. (a) Show that…

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Q: 1. Suppose a increases on [a, b], a<xo Sb, a is continuous at xo, f(xo) 1, and S(x) 0 if x xo. Prove…

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Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

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Q: a) For which values of a and b in R is f(x) continuous on (-infinity, pi/2)? b) For which values of…

A: Given function is f(x)=x+a, x≤0bcos(x)+asin(x) , 0&lt;x≤π4btan(x), x≥π4 If f(x) is continuous on…

Q: 1. Consider the function Sa(1 – x) sin r€ (0, 1), πε (0, 1) , - f (x) = %3D (0, x € {0, 1}. (a)…

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

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Q: 3. Define f:R –→ R by ƒ (x) = sin (1/x) if x # 0 and f (0) = 0. (a) Show that fis not continuous at…

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Q: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

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Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

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Q: 15. Let f, g : R → R be continuous at a point c, and let h(x) = sup (f (x). 8(x)} for x e R. Show…

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Q: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

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Q: II. Prove the following theorems on limits. 1. f(x) = L if and only if (x) – L] = 0 2. f(x) = L if…

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Q: 3. Let f, g : D →R be continuous at c € D. Prove that fg is continuous at c.

A: Since you have asked multiple questions, we have answered the first question. If you want the second…

Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

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Q: 3) Determine all points at which the following an functions are not Continuous: sin nur a) fe=で, )…

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Q: 1. Given that f is continuous on a, b|, compare f(b)(b – a) | f(x) dx, and if f is constant on a,…

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Q: 2. (i). Let g : R→R be continuous at c and let C ={r€R:f(x)=0}. If c, EC and lim c, =c. %3D Prove…

A: PART A Let g: ℝ→ℝ be continuous at c. Let C={ x∈ℝ : f(x)=0}  Let cn∈C and  lim cn=c  As,  cn∈ C…

Q: 15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show…

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Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Q: If f(x) is continuous with f(a)f(b) < 0, then the e Select one: O an indeterminable number of roots…

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