Need Real Analysis homework help. Need problems E, F, G, and H. Neat handwriting is appreciated and will receive a like.

12. Answer the questions posed in the following proof Theorem The Heierstrass pathological function is not differentiable at any real number. Proof: To simplify the proof, we will consider the special case in which a = .6 and b - 15; i.e., let Mix) - 2.6* cos (15*nx). We will argue by contradiction and assume that w is differentiable at some real number c. 8. Explain why lim{(x) - Wic))/(x. c) - L for some real number L. b. Why is there none o O auch that (N(x) - Wie)/(x - e) L + 1. Let u be the result of rounding 25'e off to the nearest integer, r- 15" - N, and X = c + (1-1)/15. In addition, let Ex(x) = .6*008 (15rx). c. Explain why 1x S .5 and x E (c-5.c+5). d. Explain why each of the following equations holds. (HX) - Wie))/x-c) - S. (Ex(x) - FX(c)]/(x-c) - H f () - De: [f,(x) - f(c)]/(x-c) + [Ex (x) - Ex{c)]/(x-c) f. x+ E c - ) - [, (x) - f1c)]/(x-c) + St. Ekz(x) - Enke)]/(x-c) e. Use The Mean Value Theorem to show | [Ex(x) - Fic) / IX-9 for all non-negative integers k. f. Show that I [(x) - f. (c)]/(x-c) )/(1-r). h[ cc) 611 /(. 1. Explain why .611 + coa (25x))/(1-r) is non-negative for all non-negative integera k and why i...6(1 + cos (15)/(1-2) 2 (1 + COB (nr>)/(1-1). j. Prove Lie (2+ cos (15 nr))/(1-2) 2 2/3 k. Why is SA (..(x) - Ele)]/(x-c) > 12/31 97 1. show that (W(x) - W(c)/1x - c) > 12/29 - 99/2 - 9/6. m. Explain why this is a contradiction 12. Answer the questions posed in the following proof Theorem The Heierstrass pathological function is not differentiable at any real number. Proof: To simplify the proof, we will consider the special case in which a = .6 and b - 15; i.e., let Mix) - 2.6* cos (15*nx). We will argue by contradiction and assume that w is differentiable at some real number c. 8. Explain why lim{(x) - Wic))/(x. c) - L for some real number L. b. Why is there none o O auch that (N(x) - Wie)/(x - e) L + 1. Let u be the result of rounding 25'e off to the nearest integer, r- 15" - N, and X = c + (1-1)/15. In addition, let Ex(x) = .6*008 (15rx). c. Explain why 1x S .5 and x E (c-5.c+5). d. Explain why each of the following equations holds. (HX) - Wie))/x-c) - S. (Ex(x) - FX(c)]/(x-c) - H f () - De: [f,(x) - f(c)]/(x-c) + [Ex (x) - Ex{c)]/(x-c) f. x+ E c - ) - [, (x) - f1c)]/(x-c) + St. Ekz(x) - Enke)]/(x-c) e. Use The Mean Value Theorem to show | [Ex(x) - Fic) / IX-9 for all non-negative integers k. f. Show that I [(x) - f. (c)]/(x-c) )/(1-r). h[ cc) 611 /(. 1. Explain why .611 + coa (25x))/(1-r) is non-negative for all non-negative integera k and why i...6(1 + cos (15)/(1-2) 2 (1 + COB (nr>)/(1-1). j. Prove Lie (2+ cos (15 nr))/(1-2) 2 2/3 k. Why is SA (..(x) - Ele)]/(x-c) > 12/31 97 1. show that (W(x) - W(c)/1x - c) > 12/29 - 99/2 - 9/6. m. Explain why this is a contradiction