1 Let {f} be a sequence of real nonnegative functions on R', and consider the following four state ments: (a) If f, and f2 are upper semicontinuous, then f, +f2 is upper semicontinuous. (b) If f, and f, are lower semicontinuous, then f, +f2 is lower semicontinuous. (c) If each f, is upper semicontinuous, then f, is upper semicontinuous. (d) If each f, is lower semicontinuous, then E f, 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 {f} be a sequence of real nonnegative functions on R', and consider the following four state ments: (a) If f, and f2 are upper semicontinuous, then f, +f2 is upper semicontinuous. (b) If f, and f, are lower semicontinuous, then f, +f2 is lower semicontinuous. (c) If each f, is upper semicontinuous, then f, is upper semicontinuous. (d) If each f, is lower semicontinuous, then E f, 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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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