Give an example of a bounded function on $[0,1]$, which doesnot achieve its infimum on any $[a,b] subset [0,1], a < b$
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Give an example of a bounded function on $[0,1]$, which doesnot achieve its infimum on any $[a,b] subset [0,1], a < b$
MY attempts :
i take beginalign
f(x)=begincases
x &text if xneq 0,1\
1/2 &textotherwise
endcases
endalign
Is its corrects ??
real-analysis
add a comment |Â
up vote
0
down vote
favorite
Give an example of a bounded function on $[0,1]$, which doesnot achieve its infimum on any $[a,b] subset [0,1], a < b$
MY attempts :
i take beginalign
f(x)=begincases
x &text if xneq 0,1\
1/2 &textotherwise
endcases
endalign
Is its corrects ??
real-analysis
1
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
thanks @TheoBendit
â stupid
Aug 18 at 5:34
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Give an example of a bounded function on $[0,1]$, which doesnot achieve its infimum on any $[a,b] subset [0,1], a < b$
MY attempts :
i take beginalign
f(x)=begincases
x &text if xneq 0,1\
1/2 &textotherwise
endcases
endalign
Is its corrects ??
real-analysis
Give an example of a bounded function on $[0,1]$, which doesnot achieve its infimum on any $[a,b] subset [0,1], a < b$
MY attempts :
i take beginalign
f(x)=begincases
x &text if xneq 0,1\
1/2 &textotherwise
endcases
endalign
Is its corrects ??
real-analysis
edited Aug 18 at 5:30
Arthur
100k793175
100k793175
asked Aug 18 at 5:16
stupid
627110
627110
1
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
thanks @TheoBendit
â stupid
Aug 18 at 5:34
add a comment |Â
1
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
thanks @TheoBendit
â stupid
Aug 18 at 5:34
1
1
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
thanks @TheoBendit
â stupid
Aug 18 at 5:34
thanks @TheoBendit
â stupid
Aug 18 at 5:34
add a comment |Â
1 Answer
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Yes. The infimum of $f$ is $0$ and the supremum is $1$, but no value of $x$ gives $0$ or $1$. So not only have you avoid the infimum, but also the supremum.
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Yes. The infimum of $f$ is $0$ and the supremum is $1$, but no value of $x$ gives $0$ or $1$. So not only have you avoid the infimum, but also the supremum.
add a comment |Â
up vote
3
down vote
accepted
Yes. The infimum of $f$ is $0$ and the supremum is $1$, but no value of $x$ gives $0$ or $1$. So not only have you avoid the infimum, but also the supremum.
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Yes. The infimum of $f$ is $0$ and the supremum is $1$, but no value of $x$ gives $0$ or $1$. So not only have you avoid the infimum, but also the supremum.
Yes. The infimum of $f$ is $0$ and the supremum is $1$, but no value of $x$ gives $0$ or $1$. So not only have you avoid the infimum, but also the supremum.
answered Aug 18 at 5:31
Arthur
100k793175
100k793175
add a comment |Â
add a comment |Â
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1
$textYes.hspace0pt$
â Theo Bendit
Aug 18 at 5:21
thanks @TheoBendit
â stupid
Aug 18 at 5:34