Let A CR be nonempty and bounded from below. Fix some real number x < 0 and {ax : a E A}. Show that sup B = x inf A. consider the set B = Suppose that f : R → R is a function such that the sets Ua = {x € R : f(x) < a} and Va = {x E R : f(x) > a} are open in R for each a E R. Show that f is continuous. %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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1. Let A CR be nonempty and bounded from below. Fix some real number x < 0 and
consider the set B =
{ax : a E A}. Show that
sup
В
= x inf A.
2. Suppose that f: R → R is a function such that the sets Ua = {x E R : f(x) < a} and
Va = {x E R : f (x) > a} are open in R for each a E R. Show that f is continuous.
3. What can you say about a set A C R, if every subset of A is closed in R?
4. Which of the following subsets of R are complete? Connected? Compact?
A = Z,
B = {x € R : a° – 3x < 4},
C = {x € R : sin x € Z}.
Transcribed Image Text:1. Let A CR be nonempty and bounded from below. Fix some real number x < 0 and consider the set B = {ax : a E A}. Show that sup В = x inf A. 2. Suppose that f: R → R is a function such that the sets Ua = {x E R : f(x) < a} and Va = {x E R : f (x) > a} are open in R for each a E R. Show that f is continuous. 3. What can you say about a set A C R, if every subset of A is closed in R? 4. Which of the following subsets of R are complete? Connected? Compact? A = Z, B = {x € R : a° – 3x < 4}, C = {x € R : sin x € Z}.
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