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1 Let {fn} bes sequence of real nonnegative functions on R', consider the following four statements: (a) If fi and f2 are upper senmicontinuous, then fi+ f2 is upper semicontin- 1uous. (b) If fi and f2 are lower semicontimuous, then fi+ f2 is upper semicontin- 1uous. (c) If each fn is upper semicontinuous, then E Sn is upper semicontinuous. (c) If cach fn is lower semicontinuous, then fn is lower semicontinuous. Show that three of these are true and that one is false. What happens if the word "nonnegative" is omitted? Is the truth of the statements affected if R is replaced by a general topological space?

1 Let {fn} bes sequence of real nonnegative functions on R', consider the following four statements: (a) If fi and f2 are upper senmicontinuous, then fi+ f2 is upper semicontin- 1uous. (b) If fi and f2 are lower semicontimuous, then fi+ f2 is upper semicontin- 1uous. (c) If each fn is upper semicontinuous, then E Sn is upper semicontinuous. (c) If cach fn is lower semicontinuous, then fn is lower semicontinuous. Show that three of these are true and that one is false. What happens if the word "nonnegative" is omitted? Is the truth of the statements affected if R is replaced by a general topological space?

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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1 Let {f.} bes sequence of real nonnegative functions on R', consider the following four statements: (a) If fi and f2 are upper semicontinuous, then fi+ f2 is upper semicontin- 11ous. (b) If fi and f2 are lower semicontimnous, then fi + f2 is upper semicontin- 1uous. (c) If each fn is upper semicontintuous, then N Sn is upper semicontinuous. (c) If cach fn is lower semicontinuous, then In is lower semicontinuous. Show that three of these are true and that one is false. What happens if the word "nonnegative" is omitted? Is the truth of the statements affected if R is replaced by a general topological space?
Transcribed Image Text:1 Let {f.} bes sequence of real nonnegative functions on R', consider the following four statements: (a) If fi and f2 are upper semicontinuous, then fi+ f2 is upper semicontin- 11ous. (b) If fi and f2 are lower semicontimnous, then fi + f2 is upper semicontin- 1uous. (c) If each fn is upper semicontintuous, then N Sn is upper semicontinuous. (c) If cach fn is lower semicontinuous, then In is lower semicontinuous. Show that three of these are true and that one is false. What happens if the word "nonnegative" is omitted? Is the truth of the statements affected if R is replaced by a general topological space?
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