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5a. Let ICR be an interval and f: I→ R be monotone. Show that every discontinuity of f is a jump discontinuity. Hint: Use Theorem 0.16 and its remark of the Lecture Notes 11. Let f: RR, be given by 5b. ƒ (x) = { x sin (1) - { 0 Show that f is not differentiable at x = 0. Hint: Need to show that the limit lim f(x)—ƒ(0) x-0 01x = if x #0 if x = 0. lim sin (1) does not exist. x-0

5a. Let ICR be an interval and f: I→ R be monotone. Show that every discontinuity of f is a jump discontinuity. Hint: Use Theorem 0.16 and its remark of the Lecture Notes 11. Let f: RR, be given by 5b. ƒ (x) = { x sin (1) - { 0 Show that f is not differentiable at x = 0. Hint: Need to show that the limit lim f(x)—ƒ(0) x-0 01x = if x #0 if x = 0. lim sin (1) does not exist. x-0

Advanced Engineering Mathematics
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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5a. Let ICR be an interval and f : I → R be monotone. Show that every discontinuity of f is a jump discontinuity. Hint: Use Theorem 0.16 and its remark of the Lecture Notes 11. Let f: R → R, be given by 5b. f(x) = { zsin(²) if x +0 0 x=0 Show that f is not differentiable at x = 0. Hint: Need to show that the limit lim f(x) —ƒ (0) x-0 0←x lim sin (1) does not exist. x →0
Transcribed Image Text:5a. Let ICR be an interval and f : I → R be monotone. Show that every discontinuity of f is a jump discontinuity. Hint: Use Theorem 0.16 and its remark of the Lecture Notes 11. Let f: R → R, be given by 5b. f(x) = { zsin(²) if x +0 0 x=0 Show that f is not differentiable at x = 0. Hint: Need to show that the limit lim f(x) —ƒ (0) x-0 0←x lim sin (1) does not exist. x →0
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