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Problem 107. Use the definition of continuity to show that Sa sin (4). ifo if -0 (r) - is continuous at 0.

Problem 107. Use the definition of continuity to show that Sa sin (4). ifo if -0 (r) - 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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11:54 ( Safari RealAnalysis-ISBN-fix.. S E o e pres tnper, e s y often as r nears zero this graph must be viewed with a certain amount of suspicion. However our completely analytic definition of continuity shows that this function is, in fact, contimuous at 0. Problem 107. Use the definition of continuity to show that S(2) - sin (2). ifz 40 10, f(2) = is continuous at 0. Even more perplexing is the function defined by Sr, if z is rational D(z) - if z is irrational. To the naked eye, the graph of this function looks like the lines y = 0 and y =z. Of course, such a graph woukd 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 x inational rkral Can such a funetion 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. contimous at 0 and at no other point. CONTINUITY: WHAT IT ISN'T AND WHAT IT Is 110 Problem 108. (a) Use the definition of continuity to show that the function Jz, ifa is rational 10, a is irrational D(r) - is continuous at 0. (b) Let a 0. Use the definition of continuity to show that D is not continuous at a. (Hint: You might want to break this up into two cases where a is rational or irrational. Show that no choice of 6 > 0 will work for e -|al. Note that Theorem Eof Chapter will probably help here.) Next 1 Dashboard Calendar To Do Notifications Inbox
Transcribed Image Text:11:54 ( Safari RealAnalysis-ISBN-fix.. S E o e pres tnper, e s y often as r nears zero this graph must be viewed with a certain amount of suspicion. However our completely analytic definition of continuity shows that this function is, in fact, contimuous at 0. Problem 107. Use the definition of continuity to show that S(2) - sin (2). ifz 40 10, f(2) = is continuous at 0. Even more perplexing is the function defined by Sr, if z is rational D(z) - if z is irrational. To the naked eye, the graph of this function looks like the lines y = 0 and y =z. Of course, such a graph woukd 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 x inational rkral Can such a funetion 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. contimous at 0 and at no other point. CONTINUITY: WHAT IT ISN'T AND WHAT IT Is 110 Problem 108. (a) Use the definition of continuity to show that the function Jz, ifa is rational 10, a is irrational D(r) - is continuous at 0. (b) Let a 0. Use the definition of continuity to show that D is not continuous at a. (Hint: You might want to break this up into two cases where a is rational or irrational. Show that no choice of 6 > 0 will work for e -|al. Note that Theorem Eof Chapter will probably help here.) Next 1 Dashboard Calendar To Do Notifications Inbox
Expert Solution
Step 1: Definition used and given.

Continuity: A function f(x) is said to be continuous at x=a, if

i) limx→af(x) exists

ii) f(a) exists and

iii) limx→af(x)=f(a)

Given: f(x)=xsin1x,if x≠00,if x=0

To show: f(x) is continuous at x=0 using definition of continuity.

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