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18.P. Give an example of an upper semi-continuous function which has an infinite number of points of discontinuity.

18.P. Give an example of an upper semi-continuous function which has an infinite number of points of discontinuity.

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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Please help with exercise 18. P

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18.L. Define what is meant by lim sup f = L, lim inf f o. 18.M. Show that if f is an upper semi-continuous function on a compact subset K of RP with values in R, then f is bounded above and attains its supremum on K. 18.N. Show that an upper semi-continuous function on a compact set may not be bounded below and may not attain its infimum. 18.0. Show that if A is an open subset of Rº and if f is defined on R to R by f(x) = 1, x E A, 0, x ¢ A, then f is a lower semi-continuous function. If A is a closed subset of RP, show that f is upper semi-continuous. 18.P. Give an example of an upper semi-continuous function which has an infinite number of points of discontinuity. 18.Q. Is it true that function on R? to R is continuous at a point if and only if it is both upper and lower semi-continuous at this point?
Transcribed Image Text:18.L. Define what is meant by lim sup f = L, lim inf f o. 18.M. Show that if f is an upper semi-continuous function on a compact subset K of RP with values in R, then f is bounded above and attains its supremum on K. 18.N. Show that an upper semi-continuous function on a compact set may not be bounded below and may not attain its infimum. 18.0. Show that if A is an open subset of Rº and if f is defined on R to R by f(x) = 1, x E A, 0, x ¢ A, then f is a lower semi-continuous function. If A is a closed subset of RP, show that f is upper semi-continuous. 18.P. Give an example of an upper semi-continuous function which has an infinite number of points of discontinuity. 18.Q. Is it true that function on R? to R is continuous at a point if and only if it is both upper and lower semi-continuous at this point?
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