Problem #3: Prove the following theorem:Theorem 2. If f: AR and c E A is an accumulation point of A, then f is continuous at c if and only iflim f(x) = f(c)8个&for every sequence (xn) in A such that xn → c as n → ∞.

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Answered: Problem #3: Prove the following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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7. If a function f is not defined at a but lim f(x) exists, is it possible for f to be continuous at a? If so, under what conditions? What kind of discontinuity must f have at a otherwise? x→a 8. If a function f is defined at a and lim f(x) exists, is it possible for f to be continuous at a? If so, under what conditions? What kind of discontinuity must f have at a otherwise? x-a 9. If lim f(x) ‡ lim f(x) but both one-sided limits exist, is it possible for f to be continuous at a? If so, x-a- x→a+ under what conditions? What kind of discontinuity must f have at a otherwise?
1. Study the continuity of f (x) at x = 2 in each of the following; 3x + 1 if x 2 x3 – 1 if x 2 if x = 2 (x+5 if x > 2 x if x<2 c) f(x) =-
(b) A function f: A → R is continuous at a point c € A if for any e > 0, unere exists such that whenever 0 0 but 8< f(x)-f(c)| ≤e+e².
Suppose f: R → R is an increasing function. (By our definitions, the words "increasing" and "nondecreasing" are synonyms, that is, "increasing" means "weakly increasing.") True or false: If x and y are real numbers, then x ≤ y if and only if f(x) ≤ f(y). True False
Theorem 6.8 Let an increasing sequence of non-negative, measur- Lim able functions (f) be defined on a set E. If n→o Jn(x) = f(x) on E, then Lim %3D
Specifically part b
1. f is continuous for a ≤ x ≤ b but not differentiable for some c such that a < c < b. Which of the following could be true? (A) x = c is a vertical asymptote of the graph of f. (D) f(c) is undefined. (A) 0 only (D) -2,2, and 6 only -2 -1 For which values of x does f'(x) = 0? (A) None (B) lim f(x) + f(c) x-c' m 1 2 3 4 (B) One (E) None of the above (B) 2 only (E) -2,0, 2, 4, and 6 (C) Two (C) The graph of f has a cusp at x = c. 5 f(x) If f'(x) = tan-¹(x²-x), at how many points is the tangent line to the graph of y = f(x) parallel to the line y 2x? (D) Three 6 7 (C) 0 and 4 only (E) Infinitely many
A function f is continuous at a number a if O a. lima+ f(x) = lima-f(x) = f(a) O b. lima+ f(x) = lima- f(x) I lima+ f(x) = f(a) lima+ f(x) = limx→a f(x) ‡ f(a) O c. O d.
2. Suppose that f(2) is an entire function and there exists a c €R s.t. Re(f(z))
For each n = 1,2, 3, ..., define fn (X) -for every x E R 1+ 2n²x? Then the function f defined by f(x) = lim fn(x) exists for each a E Rand is equal to: O a. f(x) = x O b. f(x) = x/2 O c. f(x) = x² O d. f(x) = 0 Which of the following expressions does not have an integral which can be expressed in terms of elementary functions: (1, x, In(x), sin(x), etc.) O a. ez? O b. sin(x)e* 1 O d. ez

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Problem #3: Prove the following theorem: Theorem 2. If f: AR and c E A is an accumulation point of A, then f is continuous at c if and only if lim f(x) = f(c) 8个& for every sequence (xn) in A such that xn → c as n → ∞.