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Problem 9.1.19. The Topólógist's Sinė Function. Use the definition of continuity to show that Sæ sin (), if x +0 f(x) = if x = 0 %3D is continuous at 0.

Problem 9.1.19. The Topólógist's Sinė Function. Use the definition of continuity to show that Sæ sin (), if x +0 f(x) = if x = 0 %3D is continuous at 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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12:17 M A personal.psu.edu/ecb! + To the naked eye, the graph of this function looks like the lines y = x. Of course, such a graph would not be the graph of a function. Actually, both 0 and of these lines have holes in them. Wherever there is a point on one line there is a "hole" on the other. Each of these holes are the width of a single point (that is, their "width" is zero!) so they are invisible to the naked eye (or even magnified under the most powerful microscope available). This idea is illustrated in the following graph 0.5- -0.5 0.5 x irrationa) x rational Can such a function so "full of holes" actually be continuous anywhere? It turns out that we can use our definition to show that this function is, in fact, continuous at 0 and at no other point. II II
Transcribed Image Text:12:17 M A personal.psu.edu/ecb! + To the naked eye, the graph of this function looks like the lines y = x. Of course, such a graph would not be the graph of a function. Actually, both 0 and of these lines have holes in them. Wherever there is a point on one line there is a "hole" on the other. Each of these holes are the width of a single point (that is, their "width" is zero!) so they are invisible to the naked eye (or even magnified under the most powerful microscope available). This idea is illustrated in the following graph 0.5- -0.5 0.5 x irrationa) x rational Can such a function so "full of holes" actually be continuous anywhere? It turns out that we can use our definition to show that this function is, in fact, continuous at 0 and at no other point. II II
12:17 i Tact, continuous a U. Problem 9.1.19. The Topologist's Sine Function. Use the definition of continuity to show that x sin (-), ifx # 0 f(x) = {. if x = 0 is continuous at 0. Even more perplexing is the function defined by Sæ, if x is rational D(x) = 0, 10, if x is irrational . To the naked eye, the graph of this function looks like the lines y = 0 and y = x. Of course, such a graph would not be the graph of a function. Actually, both of these lines have holes in them. Wherever there is a point on one line there is a "hole" on the other. Each of these holes are the width of a single point (that is, their "width" is zero!) so they are invisible to the naked eye (or even magnified under the most powerful microscope available). This idea is illustrated in the following graph II II
Transcribed Image Text:12:17 i Tact, continuous a U. Problem 9.1.19. The Topologist's Sine Function. Use the definition of continuity to show that x sin (-), ifx # 0 f(x) = {. if x = 0 is continuous at 0. Even more perplexing is the function defined by Sæ, if x is rational D(x) = 0, 10, if x is irrational . To the naked eye, the graph of this function looks like the lines y = 0 and y = x. Of course, such a graph would not be the graph of a function. Actually, both of these lines have holes in them. Wherever there is a point on one line there is a "hole" on the other. Each of these holes are the width of a single point (that is, their "width" is zero!) so they are invisible to the naked eye (or even magnified under the most powerful microscope available). This idea is illustrated in the following graph II II
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