Multivariable calculus discontinuities question
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Question: If D is the set of discontinuities of $f:mathbb{R}^{n}rightarrow mathbb{R}$, show that the set of discontinuities of $f_{Phi_{A}}$ is contained in $D bigcup partial A$.
So, $f_{Phi_{A}}$ is the characteristic function of $f$, which is begin{cases}
1 & if xin A \
0 & if xnotin A \
end{cases}
Then, pick a point $pin f_{Phi_{A}}$ and show that $p$ is contained in $D bigcup partial A$.
Is this the right way to prove this? Any hints would be appreciated. Thank you.
calculus real-analysis multivariable-calculus characteristic-functions discontinuous-functions
asked Nov 26 at 9:53
hard
83
add a comment |
up vote
-1
down vote
favorite
Question: If D is the set of discontinuities of $f:mathbb{R}^{n}rightarrow mathbb{R}$, show that the set of discontinuities of $f_{Phi_{A}}$ is contained in $D bigcup partial A$.
So, $f_{Phi_{A}}$ is the characteristic function of $f$, which is begin{cases}
1 & if xin A \
0 & if xnotin A \
end{cases}
Then, pick a point $pin f_{Phi_{A}}$ and show that $p$ is contained in $D bigcup partial A$.
Is this the right way to prove this? Any hints would be appreciated. Thank you.
calculus real-analysis multivariable-calculus characteristic-functions discontinuous-functions
asked Nov 26 at 9:53
hard
83
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Question: If D is the set of discontinuities of $f:mathbb{R}^{n}rightarrow mathbb{R}$, show that the set of discontinuities of $f_{Phi_{A}}$ is contained in $D bigcup partial A$.
So, $f_{Phi_{A}}$ is the characteristic function of $f$, which is begin{cases}
1 & if xin A \
0 & if xnotin A \
end{cases}
Then, pick a point $pin f_{Phi_{A}}$ and show that $p$ is contained in $D bigcup partial A$.
Is this the right way to prove this? Any hints would be appreciated. Thank you.
calculus real-analysis multivariable-calculus characteristic-functions discontinuous-functions
asked Nov 26 at 9:53
hard
83
Question: If D is the set of discontinuities of $f:mathbb{R}^{n}rightarrow mathbb{R}$, show that the set of discontinuities of $f_{Phi_{A}}$ is contained in $D bigcup partial A$.
So, $f_{Phi_{A}}$ is the characteristic function of $f$, which is begin{cases}
1 & if xin A \
0 & if xnotin A \
end{cases}
Then, pick a point $pin f_{Phi_{A}}$ and show that $p$ is contained in $D bigcup partial A$.
Is this the right way to prove this? Any hints would be appreciated. Thank you.
calculus real-analysis multivariable-calculus characteristic-functions discontinuous-functions
calculus real-analysis multivariable-calculus characteristic-functions discontinuous-functions
asked Nov 26 at 9:53
hard
83
asked Nov 26 at 9:53
hard
83
edited Nov 26 at 9:59
asked Nov 26 at 9:53
hard
83
asked Nov 26 at 9:53
hard
83
asked Nov 26 at 9:53
hard
83
83
add a comment |
add a comment |
1 Answer
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What you have to show is $fPhi $ is continuous at $x$ if $x notin Dcup partial A$. So assume $x notin Dcup partial A$ and take $x_n to x$. Since $x notin partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(fPhi )(x_n)=f (x_n) to f(x) =(fPhi ) (x)$ because $x notin D$. Similar argument in the second case.
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
What you have to show is $fPhi $ is continuous at $x$ if $x notin Dcup partial A$. So assume $x notin Dcup partial A$ and take $x_n to x$. Since $x notin partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(fPhi )(x_n)=f (x_n) to f(x) =(fPhi ) (x)$ because $x notin D$. Similar argument in the second case.
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
add a comment |
up vote
1
down vote
accepted
What you have to show is $fPhi $ is continuous at $x$ if $x notin Dcup partial A$. So assume $x notin Dcup partial A$ and take $x_n to x$. Since $x notin partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(fPhi )(x_n)=f (x_n) to f(x) =(fPhi ) (x)$ because $x notin D$. Similar argument in the second case.
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
What you have to show is $fPhi $ is continuous at $x$ if $x notin Dcup partial A$. So assume $x notin Dcup partial A$ and take $x_n to x$. Since $x notin partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(fPhi )(x_n)=f (x_n) to f(x) =(fPhi ) (x)$ because $x notin D$. Similar argument in the second case.
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
What you have to show is $fPhi $ is continuous at $x$ if $x notin Dcup partial A$. So assume $x notin Dcup partial A$ and take $x_n to x$. Since $x notin partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(fPhi )(x_n)=f (x_n) to f(x) =(fPhi ) (x)$ because $x notin D$. Similar argument in the second case.
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
answered Nov 26 at 10:00
Kavi Rama Murthy
45.4k31852
45.4k31852
add a comment |
add a comment |
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