Example of a function f and a set E with the following: f is uniformly continuous on E, but f doesn't attain either a max or a min on E.Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max.Is there an unbounded uniformly continuous function with a bounded domain?Determine if the following function is continuous in $(0,0)$.continuous map of connected set is connected, example: Proving the connectedness of this set.Min/max of a continuous functionWhat is an example of a uniformly continuous function but not absolutely continuousA function continuous and bounded on a closed and bounded set but not uniformly continuous thereGive an example of a function which is continuous on $mathbbQcap[2,4]$ but not uniformly continuous on the same set.the function is continuous on which setShow the existence of a global maximum of a continuous function with unbounded domain
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Example of a function f and a set E with the following: f is uniformly continuous on E, but f doesn't attain either a max or a min on E.
Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max.Is there an unbounded uniformly continuous function with a bounded domain?Determine if the following function is continuous in $(0,0)$.continuous map of connected set is connected, example: Proving the connectedness of this set.Min/max of a continuous functionWhat is an example of a uniformly continuous function but not absolutely continuousA function continuous and bounded on a closed and bounded set but not uniformly continuous thereGive an example of a function which is continuous on $mathbbQcap[2,4]$ but not uniformly continuous on the same set.the function is continuous on which setShow the existence of a global maximum of a continuous function with unbounded domain
$begingroup$
I thought I could use a constant function but I think absolute maximums are attained on a constant function.
continuity extreme-value-theorem
asked Mar 28 at 20:31
MD3MD3
412
$endgroup$
add a comment |
$begingroup$
I thought I could use a constant function but I think absolute maximums are attained on a constant function.
continuity extreme-value-theorem
asked Mar 28 at 20:31
MD3MD3
412
$endgroup$
add a comment |
$begingroup$
I thought I could use a constant function but I think absolute maximums are attained on a constant function.
continuity extreme-value-theorem
asked Mar 28 at 20:31
MD3MD3
412
$endgroup$
I thought I could use a constant function but I think absolute maximums are attained on a constant function.
continuity extreme-value-theorem
continuity extreme-value-theorem
asked Mar 28 at 20:31
MD3MD3
412
asked Mar 28 at 20:31
MD3MD3
412
asked Mar 28 at 20:31
MD3MD3
412
asked Mar 28 at 20:31
MD3MD3
412
asked Mar 28 at 20:31
MD3MD3
412
412
add a comment |
add a comment |
2 Answers
2
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Consider $f:mathbbRtomathbbR$ with $f(x):=x$. Since for any $x,yinmathbbR$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.
answered Mar 28 at 20:47
MaksimMaksim
98719
$endgroup$
add a comment |
$begingroup$
The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?
answered Mar 28 at 20:35
jmerryjmerry
17k11633
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2 Answers
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$begingroup$
Consider $f:mathbbRtomathbbR$ with $f(x):=x$. Since for any $x,yinmathbbR$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.
answered Mar 28 at 20:47
MaksimMaksim
98719
$endgroup$
add a comment |
$begingroup$
Consider $f:mathbbRtomathbbR$ with $f(x):=x$. Since for any $x,yinmathbbR$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.
answered Mar 28 at 20:47
MaksimMaksim
98719
$endgroup$
add a comment |
$begingroup$
Consider $f:mathbbRtomathbbR$ with $f(x):=x$. Since for any $x,yinmathbbR$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.
answered Mar 28 at 20:47
MaksimMaksim
98719
$endgroup$
Consider $f:mathbbRtomathbbR$ with $f(x):=x$. Since for any $x,yinmathbbR$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.
answered Mar 28 at 20:47
MaksimMaksim
98719
answered Mar 28 at 20:47
MaksimMaksim
98719
answered Mar 28 at 20:47
MaksimMaksim
98719
answered Mar 28 at 20:47
MaksimMaksim
98719
98719
add a comment |
add a comment |
$begingroup$
The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?
answered Mar 28 at 20:35
jmerryjmerry
17k11633
$endgroup$
add a comment |
$begingroup$
The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?
answered Mar 28 at 20:35
jmerryjmerry
17k11633
$endgroup$
add a comment |
$begingroup$
The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?
answered Mar 28 at 20:35
jmerryjmerry
17k11633
$endgroup$
The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?
answered Mar 28 at 20:35
jmerryjmerry
17k11633
answered Mar 28 at 20:35
jmerryjmerry
17k11633
answered Mar 28 at 20:35
jmerryjmerry
17k11633
answered Mar 28 at 20:35
jmerryjmerry
17k11633
17k11633
add a comment |
add a comment |
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