Exercise 2. Let f: [0, 1] → [0, 1] be the function given by [ 2x, if z € [0, 1] f(x) = { 2-2r, if x €, 1] a) Make a rough sketch of the graph of f. b) Show that f is continuous at a, for every point a € [0, 1].
Exercise 2. Let f: [0, 1] → [0, 1] be the function given by [ 2x, if z € [0, 1] f(x) = { 2-2r, if x €, 1] a) Make a rough sketch of the graph of f. b) Show that f is continuous at a, for every point a € [0, 1].
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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![Exercise 2. Let f: [0, 1] → [0, 1] be the function given by
if z €
f(x) = { 2 - 2x, if a € [1,1]
a) Make a rough sketch of the graph of f.
b) Show that f is continuous at a, for every point a € [0, 1].
c) Find a periodic point ₺ € [0, 1] of period three.
d) Find a periodic point c € [0, 1] of period five.
e) Look into the mathematical literature and find Sharkovskii's theorem. Formally state a simplified
version of Sharkovskii's theorem and use that to show that c) implies d).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Fcb5e2d4c-3663-4c79-9810-2de6e0a7cd65%2Fzs0nky5_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 2. Let f: [0, 1] → [0, 1] be the function given by
if z €
f(x) = { 2 - 2x, if a € [1,1]
a) Make a rough sketch of the graph of f.
b) Show that f is continuous at a, for every point a € [0, 1].
c) Find a periodic point ₺ € [0, 1] of period three.
d) Find a periodic point c € [0, 1] of period five.
e) Look into the mathematical literature and find Sharkovskii's theorem. Formally state a simplified
version of Sharkovskii's theorem and use that to show that c) implies d).
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