Oo(t) -{ = [1, if t = ² = Q and GCD(p, q) = 1, 0, if t & Q, where GCD(p, q) is the greatest common divisor of p and q. Note that 0(m) = 1, for every m Z and 0o (p/2") = 1/2", for all integers p E Z odd and n € N. Show that is continuous at each point of R \ Q and not continuous at any point of Q.
Oo(t) -{ = [1, if t = ² = Q and GCD(p, q) = 1, 0, if t & Q, where GCD(p, q) is the greatest common divisor of p and q. Note that 0(m) = 1, for every m Z and 0o (p/2") = 1/2", for all integers p E Z odd and n € N. Show that is continuous at each point of R \ Q and not continuous at any point of Q.
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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Question
#6
![Exercise 6 (Thomae). The Thomae function 0: RR is defined by
[², if t = = Q and GCD(p, q) = 1,
9
0, ift & Q,
Oo(t)
=
where GCD(p, q) is the greatest common divisor of p and q.
Note that 0(m) = 1, for every mE Z and 0o (p/2") 1/2", for all integers p E Z odd and n € N.
Show that is continuous at each point of R Q and not continuous at any point of Q.
=
Point plot of 0 on the interval [0, 1]. The topmost point in the middle shows (1/2) = 1/2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F78286dff-37f8-4e23-9c8e-a6b4046dbda2%2Fjotzas_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 6 (Thomae). The Thomae function 0: RR is defined by
[², if t = = Q and GCD(p, q) = 1,
9
0, ift & Q,
Oo(t)
=
where GCD(p, q) is the greatest common divisor of p and q.
Note that 0(m) = 1, for every mE Z and 0o (p/2") 1/2", for all integers p E Z odd and n € N.
Show that is continuous at each point of R Q and not continuous at any point of Q.
=
Point plot of 0 on the interval [0, 1]. The topmost point in the middle shows (1/2) = 1/2.
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