A question about the discontinuties and countable sets.
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Let $f$ be an arbitary real function on $mathbb{R}$. Let $A$ denote the set of left continuous points of $f$, and $B$ the set of discontinuous points of $f$. Show that $Abackslash B$ is countable.
I know the jumps of $f$ are countable, but my question seems more hard (=o=)...
real-analysis
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
add a comment |
up vote
-1
down vote
favorite
Let $f$ be an arbitary real function on $mathbb{R}$. Let $A$ denote the set of left continuous points of $f$, and $B$ the set of discontinuous points of $f$. Show that $Abackslash B$ is countable.
I know the jumps of $f$ are countable, but my question seems more hard (=o=)...
real-analysis
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
1
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $f$ be an arbitary real function on $mathbb{R}$. Let $A$ denote the set of left continuous points of $f$, and $B$ the set of discontinuous points of $f$. Show that $Abackslash B$ is countable.
I know the jumps of $f$ are countable, but my question seems more hard (=o=)...
real-analysis
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
Let $f$ be an arbitary real function on $mathbb{R}$. Let $A$ denote the set of left continuous points of $f$, and $B$ the set of discontinuous points of $f$. Show that $Abackslash B$ is countable.
I know the jumps of $f$ are countable, but my question seems more hard (=o=)...
real-analysis
real-analysis
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
edited Nov 28 at 9:59
Asaf Karagila♦
301k32422753
301k32422753
asked Nov 28 at 9:36
ChaosInferno
1
asked Nov 28 at 9:36
ChaosInferno
1
asked Nov 28 at 9:36
ChaosInferno
1
1
1
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02
add a comment |
1
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02
1
1
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02
add a comment |
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1
if $f$ is contiunuous then $A=mathbb R$ and $B$ is empty so $Asetminus B=mathbb R$
– user126154
Nov 28 at 9:44
if $f$ is the dirichelet function then $A$ is empty and $B=mathbb R$ so $Bsetminus A=mathbb R$
– user126154
Nov 28 at 9:45
maybe you want $Bcap A$. In this case you can find an anwer here math.stackexchange.com/questions/65941/…
– user126154
Nov 28 at 9:48
@user126154. The answer to the Q in your link is very nice.
– DanielWainfleet
Nov 28 at 13:02