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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?

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?

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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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? 18.R. If (fn) is a bounded sequence of continuous functions on R? to R and if f* is defined on Rº by * (x) , = sup {fn(x):n e N}, x € R?, then is it true that f* is upper semi-continuous on RP? 18.S. If (fa) is a bounded sequence of continuous functions on Rº to R and if fa is defined on R by f+ (x) = inf {fm (x):n € N}, x € RP, then is it true that fy is upper semi-continuous on R»?
Transcribed Image Text: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? 18.R. If (fn) is a bounded sequence of continuous functions on R? to R and if f* is defined on Rº by * (x) , = sup {fn(x):n e N}, x € R?, then is it true that f* is upper semi-continuous on RP? 18.S. If (fa) is a bounded sequence of continuous functions on Rº to R and if fa is defined on R by f+ (x) = inf {fm (x):n € N}, x € RP, then is it true that fy is upper semi-continuous on R»?
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