Topology: continuity on product of (metric) spaces












1












$begingroup$


What are the conditions so that the function defined on the product space $Xtimes Y$



$f: X times Yrightarrow mathbb{Z}$ is continuous. For example, is there a condition that says that if any restriction on $X times{y_0}$ or ${x_0}times Y $ is continuous, then $f$ is continuous? Are there other conditions if we deal with metric spaces ?






general-topology continuity







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$endgroup$












  • $begingroup$
    Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:44


















1












$begingroup$


What are the conditions so that the function defined on the product space $Xtimes Y$



$f: X times Yrightarrow mathbb{Z}$ is continuous. For example, is there a condition that says that if any restriction on $X times{y_0}$ or ${x_0}times Y $ is continuous, then $f$ is continuous? Are there other conditions if we deal with metric spaces ?






general-topology continuity







share|cite|improve this question











$endgroup$












  • $begingroup$
    Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:44
















1












1








1





$begingroup$


What are the conditions so that the function defined on the product space $Xtimes Y$



$f: X times Yrightarrow mathbb{Z}$ is continuous. For example, is there a condition that says that if any restriction on $X times{y_0}$ or ${x_0}times Y $ is continuous, then $f$ is continuous? Are there other conditions if we deal with metric spaces ?






general-topology continuity







share|cite|improve this question











$endgroup$




What are the conditions so that the function defined on the product space $Xtimes Y$



$f: X times Yrightarrow mathbb{Z}$ is continuous. For example, is there a condition that says that if any restriction on $X times{y_0}$ or ${x_0}times Y $ is continuous, then $f$ is continuous? Are there other conditions if we deal with metric spaces ?






general-topology continuity




general-topology continuity






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share|cite|improve this question













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edited Jan 3 at 12:32









José Carlos Santos

172k23132240




172k23132240










asked Jul 1 '18 at 16:49









PaoPao

506




506












  • $begingroup$
    Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:44




















  • $begingroup$
    Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:44


















$begingroup$
Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
$endgroup$
– DanielWainfleet
Jul 1 '18 at 17:44






$begingroup$
Part of the def'n of a Topological Group $G$ is that $f:Gtimes Gto G$, where $f(a,b)=ab$, is continuous. I have an example of a group where $f(a,b)=ab$ is continuous in each variable but $ f$ is not continuous....If a topological group is a $T_0$ space then it is a $T_n$ space for $nleq 3frac {1}{2}.$... My example (which is NOT a Top'l Group although it is a topology ON a group), is $ T_1$ but not $ T_2$
$endgroup$
– DanielWainfleet
Jul 1 '18 at 17:44












1 Answer
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$begingroup$

There is no such a simple condition as that. Consider the map$$begin{array}{rccc}fcolon&mathbb{R}^2&longrightarrow&mathbb R\&(x,y)&mapsto&begin{cases}frac{xy}{x^2+y^2}&text{ if }(x,y)neq(0,0)\0&text{ otherwise.}end{cases}end{array}$$Then $f$ is discontinuous, but each map $xmapsto f(x,y_0)$ and $ymapsto f(x_0,y)$ is continuous.






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  • 1




    $begingroup$
    Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:33














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1 Answer
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active

oldest

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1 Answer
1






active

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active

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active

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4












$begingroup$

There is no such a simple condition as that. Consider the map$$begin{array}{rccc}fcolon&mathbb{R}^2&longrightarrow&mathbb R\&(x,y)&mapsto&begin{cases}frac{xy}{x^2+y^2}&text{ if }(x,y)neq(0,0)\0&text{ otherwise.}end{cases}end{array}$$Then $f$ is discontinuous, but each map $xmapsto f(x,y_0)$ and $ymapsto f(x_0,y)$ is continuous.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:33


















4












$begingroup$

There is no such a simple condition as that. Consider the map$$begin{array}{rccc}fcolon&mathbb{R}^2&longrightarrow&mathbb R\&(x,y)&mapsto&begin{cases}frac{xy}{x^2+y^2}&text{ if }(x,y)neq(0,0)\0&text{ otherwise.}end{cases}end{array}$$Then $f$ is discontinuous, but each map $xmapsto f(x,y_0)$ and $ymapsto f(x_0,y)$ is continuous.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:33
















4












4








4





$begingroup$

There is no such a simple condition as that. Consider the map$$begin{array}{rccc}fcolon&mathbb{R}^2&longrightarrow&mathbb R\&(x,y)&mapsto&begin{cases}frac{xy}{x^2+y^2}&text{ if }(x,y)neq(0,0)\0&text{ otherwise.}end{cases}end{array}$$Then $f$ is discontinuous, but each map $xmapsto f(x,y_0)$ and $ymapsto f(x_0,y)$ is continuous.






share|cite|improve this answer









$endgroup$



There is no such a simple condition as that. Consider the map$$begin{array}{rccc}fcolon&mathbb{R}^2&longrightarrow&mathbb R\&(x,y)&mapsto&begin{cases}frac{xy}{x^2+y^2}&text{ if }(x,y)neq(0,0)\0&text{ otherwise.}end{cases}end{array}$$Then $f$ is discontinuous, but each map $xmapsto f(x,y_0)$ and $ymapsto f(x_0,y)$ is continuous.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jul 1 '18 at 17:17









José Carlos SantosJosé Carlos Santos

172k23132240




172k23132240








  • 1




    $begingroup$
    Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:33
















  • 1




    $begingroup$
    Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
    $endgroup$
    – DanielWainfleet
    Jul 1 '18 at 17:33










1




1




$begingroup$
Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
$endgroup$
– DanielWainfleet
Jul 1 '18 at 17:33






$begingroup$
Even the partial derivatives exist everywhere, but $f$ is still discontinuous at $(0,0).$............+1
$endgroup$
– DanielWainfleet
Jul 1 '18 at 17:33




















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