Does Density Property assure the continuity of a Non-Contractive Function?
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I want to be sure about a detail:
We know that, any contractive function is continuous. Reformulated in other words, any function that does not increase distance is continuous.
But, suppose we are in $mathbb{R}$, and we have a distance increasing function $f$ on a subset $A$ of $mathbb{R}$. Does the density property of $A$ in $mathbb{R}$ guarantee the continuity of this function $f$? ie. Will the image of $A$ by $f$ still be dense in $mathbb{R}$?
general-topology continuity metric-spaces
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
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add a comment |
$begingroup$
I want to be sure about a detail:
We know that, any contractive function is continuous. Reformulated in other words, any function that does not increase distance is continuous.
But, suppose we are in $mathbb{R}$, and we have a distance increasing function $f$ on a subset $A$ of $mathbb{R}$. Does the density property of $A$ in $mathbb{R}$ guarantee the continuity of this function $f$? ie. Will the image of $A$ by $f$ still be dense in $mathbb{R}$?
general-topology continuity metric-spaces
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
$endgroup$
add a comment |
$begingroup$
I want to be sure about a detail:
We know that, any contractive function is continuous. Reformulated in other words, any function that does not increase distance is continuous.
But, suppose we are in $mathbb{R}$, and we have a distance increasing function $f$ on a subset $A$ of $mathbb{R}$. Does the density property of $A$ in $mathbb{R}$ guarantee the continuity of this function $f$? ie. Will the image of $A$ by $f$ still be dense in $mathbb{R}$?
general-topology continuity metric-spaces
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
$endgroup$
I want to be sure about a detail:
We know that, any contractive function is continuous. Reformulated in other words, any function that does not increase distance is continuous.
But, suppose we are in $mathbb{R}$, and we have a distance increasing function $f$ on a subset $A$ of $mathbb{R}$. Does the density property of $A$ in $mathbb{R}$ guarantee the continuity of this function $f$? ie. Will the image of $A$ by $f$ still be dense in $mathbb{R}$?
general-topology continuity metric-spaces
general-topology continuity metric-spaces
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
edited Jan 4 at 23:30
freehumorist
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
asked Jan 4 at 23:23
freehumoristfreehumorist
351214
351214
add a comment |
add a comment |
1 Answer
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No, you can define $f: mathbb{R} to mathbb{R}$ by $f(x) = frac12 x$ for $x in mathbb{Q}$ and $f(x) = 2x$ for $x notin mathbb{Q}$. $f$ is contractive on the dense set $mathbb{Q}$ but not on $mathbb{R}$ and nowhere continuous except at $x=0$.
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
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$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
|
show 2 more comments
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
No, you can define $f: mathbb{R} to mathbb{R}$ by $f(x) = frac12 x$ for $x in mathbb{Q}$ and $f(x) = 2x$ for $x notin mathbb{Q}$. $f$ is contractive on the dense set $mathbb{Q}$ but not on $mathbb{R}$ and nowhere continuous except at $x=0$.
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
$endgroup$
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
|
show 2 more comments
$begingroup$
No, you can define $f: mathbb{R} to mathbb{R}$ by $f(x) = frac12 x$ for $x in mathbb{Q}$ and $f(x) = 2x$ for $x notin mathbb{Q}$. $f$ is contractive on the dense set $mathbb{Q}$ but not on $mathbb{R}$ and nowhere continuous except at $x=0$.
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
$endgroup$
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
|
show 2 more comments
$begingroup$
No, you can define $f: mathbb{R} to mathbb{R}$ by $f(x) = frac12 x$ for $x in mathbb{Q}$ and $f(x) = 2x$ for $x notin mathbb{Q}$. $f$ is contractive on the dense set $mathbb{Q}$ but not on $mathbb{R}$ and nowhere continuous except at $x=0$.
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
$endgroup$
No, you can define $f: mathbb{R} to mathbb{R}$ by $f(x) = frac12 x$ for $x in mathbb{Q}$ and $f(x) = 2x$ for $x notin mathbb{Q}$. $f$ is contractive on the dense set $mathbb{Q}$ but not on $mathbb{R}$ and nowhere continuous except at $x=0$.
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
answered Jan 4 at 23:43
Henno BrandsmaHenno Brandsma
112k348121
112k348121
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
|
show 2 more comments
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
you saying there exists non continuous functions which are contracting too, right?
$endgroup$
– freehumorist
Jan 4 at 23:49
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
@freehumorist No, this function is not contracting and it's not continuous.
$endgroup$
– Henno Brandsma
Jan 4 at 23:50
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
so how is this an answer to my question? I thought I asked that, when we have a distance-increasing function over a dense set, the continuity would be preserved in the image set too...
$endgroup$
– freehumorist
Jan 4 at 23:53
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
@freehumorist we also have here a distance increasing function on the dense irrationals and no continuity. So it does answer that too. BTW Distance increasingness does not imply continuity, decreasingness does (but on the whole domain).
$endgroup$
– Henno Brandsma
Jan 4 at 23:55
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
$begingroup$
Ok I got your answer. I think I should add another criteria: I want the function to be linear and monotone. Your example amazed me, now that I get; though it is not the answer I wanted.
$endgroup$
– freehumorist
Jan 4 at 23:59
|
show 2 more comments
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