Topology: continuity on product of (metric) spaces
$begingroup$
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
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
$endgroup$
add a comment |
$begingroup$
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
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
$endgroup$
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
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
edited Jan 3 at 12:32
José Carlos Santos
172k23132240
172k23132240
asked Jul 1 '18 at 16:49
PaoPao
506
asked Jul 1 '18 at 16:49
PaoPao
506
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
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oldest
votes
$begingroup$
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.
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
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
add a comment |
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1 Answer
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active
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1 Answer
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active
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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.
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
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
add a comment |
$begingroup$
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.
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
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
add a comment |
$begingroup$
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.
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
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.
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
answered Jul 1 '18 at 17:17
José Carlos SantosJosé Carlos Santos
172k23132240
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
add a comment |
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
add a comment |
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$begingroup$
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