A question about the discontinuties and countable sets.












-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=)...










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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
















-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=)...










share|cite|improve this question




















  • 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








-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=)...










share|cite|improve this question















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






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 at 9:59









Asaf Karagila

301k32422754




301k32422754










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














  • 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















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