Mixing
Proving a function is continuous by writing it as composition of continuous functions
$begingroup$
Given the following functions
$$ f:R^3 rightarrow R^2, hspace{20pt} f(x_1,x_2,x_3) = ((x_1)^2+x_1x_3, (x_2)^2+x_1x_3)$$
$$g:R^9 rightarrow R, hspace{20pt} g(A)=|x_{11} + x_{22} + x_{33}| text{where A is real} 3times3 text{matrix}$$
I need to prove that they are continuous w.r.t the corresponding euclidean metric by writing them as composition of continuous functions. Problem is I find it very confusing when it's a map between different dimensions so not sure how those functions need to look.
general-topology metric-spaces
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
$endgroup$
add a comment |
$begingroup$
Given the following functions
$$ f:R^3 rightarrow R^2, hspace{20pt} f(x_1,x_2,x_3) = ((x_1)^2+x_1x_3, (x_2)^2+x_1x_3)$$
$$g:R^9 rightarrow R, hspace{20pt} g(A)=|x_{11} + x_{22} + x_{33}| text{where A is real} 3times3 text{matrix}$$
I need to prove that they are continuous w.r.t the corresponding euclidean metric by writing them as composition of continuous functions. Problem is I find it very confusing when it's a map between different dimensions so not sure how those functions need to look.
general-topology metric-spaces
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
$endgroup$
add a comment |
$begingroup$
Given the following functions
$$ f:R^3 rightarrow R^2, hspace{20pt} f(x_1,x_2,x_3) = ((x_1)^2+x_1x_3, (x_2)^2+x_1x_3)$$
$$g:R^9 rightarrow R, hspace{20pt} g(A)=|x_{11} + x_{22} + x_{33}| text{where A is real} 3times3 text{matrix}$$
I need to prove that they are continuous w.r.t the corresponding euclidean metric by writing them as composition of continuous functions. Problem is I find it very confusing when it's a map between different dimensions so not sure how those functions need to look.
general-topology metric-spaces
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
$endgroup$
Given the following functions
$$ f:R^3 rightarrow R^2, hspace{20pt} f(x_1,x_2,x_3) = ((x_1)^2+x_1x_3, (x_2)^2+x_1x_3)$$
$$g:R^9 rightarrow R, hspace{20pt} g(A)=|x_{11} + x_{22} + x_{33}| text{where A is real} 3times3 text{matrix}$$
I need to prove that they are continuous w.r.t the corresponding euclidean metric by writing them as composition of continuous functions. Problem is I find it very confusing when it's a map between different dimensions so not sure how those functions need to look.
general-topology metric-spaces
general-topology metric-spaces
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
asked Jan 14 at 16:19
DreaDkDreaDk
6361318
6361318
add a comment |
add a comment |
1 Answer
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$begingroup$
You need:
- Projections $p_i:xmapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
- If you have 2 continuous functions $f,g:Rto R$ then $(f,g):R^2to R^2; (x,y)mapsto (f(x),g(y))$ is continuous.
- Also $+:R^2 to R; (x,y)mapsto x+y$ and $M:R^2 to R; (x,y)mapsto xcdot y$ are continuous.
So you can define (spoiler):
$f(x_1,x_2,x_3)= (+(Mcirc(p_1,p_1),Mcirc(p_1,p_3)), +(Mcirc(p_2,p_2),Mcirc(p_1,p_3)) ) (x_1,x_2,x_3)$
For g you also need the absolute value function $|cdot|$ to be continuous.
answered Jan 14 at 16:42
Felix B.Felix B.
739217
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add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
You need:
- Projections $p_i:xmapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
- If you have 2 continuous functions $f,g:Rto R$ then $(f,g):R^2to R^2; (x,y)mapsto (f(x),g(y))$ is continuous.
- Also $+:R^2 to R; (x,y)mapsto x+y$ and $M:R^2 to R; (x,y)mapsto xcdot y$ are continuous.
So you can define (spoiler):
$f(x_1,x_2,x_3)= (+(Mcirc(p_1,p_1),Mcirc(p_1,p_3)), +(Mcirc(p_2,p_2),Mcirc(p_1,p_3)) ) (x_1,x_2,x_3)$
For g you also need the absolute value function $|cdot|$ to be continuous.
answered Jan 14 at 16:42
Felix B.Felix B.
739217
$endgroup$
add a comment |
$begingroup$
You need:
- Projections $p_i:xmapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
- If you have 2 continuous functions $f,g:Rto R$ then $(f,g):R^2to R^2; (x,y)mapsto (f(x),g(y))$ is continuous.
- Also $+:R^2 to R; (x,y)mapsto x+y$ and $M:R^2 to R; (x,y)mapsto xcdot y$ are continuous.
So you can define (spoiler):
$f(x_1,x_2,x_3)= (+(Mcirc(p_1,p_1),Mcirc(p_1,p_3)), +(Mcirc(p_2,p_2),Mcirc(p_1,p_3)) ) (x_1,x_2,x_3)$
For g you also need the absolute value function $|cdot|$ to be continuous.
answered Jan 14 at 16:42
Felix B.Felix B.
739217
$endgroup$
add a comment |
$begingroup$
You need:
- Projections $p_i:xmapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
- If you have 2 continuous functions $f,g:Rto R$ then $(f,g):R^2to R^2; (x,y)mapsto (f(x),g(y))$ is continuous.
- Also $+:R^2 to R; (x,y)mapsto x+y$ and $M:R^2 to R; (x,y)mapsto xcdot y$ are continuous.
So you can define (spoiler):
$f(x_1,x_2,x_3)= (+(Mcirc(p_1,p_1),Mcirc(p_1,p_3)), +(Mcirc(p_2,p_2),Mcirc(p_1,p_3)) ) (x_1,x_2,x_3)$
For g you also need the absolute value function $|cdot|$ to be continuous.
answered Jan 14 at 16:42
Felix B.Felix B.
739217
$endgroup$
You need:
- Projections $p_i:xmapsto x_i$ are continuous. Which is how you get components (decrease dimensions).
- If you have 2 continuous functions $f,g:Rto R$ then $(f,g):R^2to R^2; (x,y)mapsto (f(x),g(y))$ is continuous.
- Also $+:R^2 to R; (x,y)mapsto x+y$ and $M:R^2 to R; (x,y)mapsto xcdot y$ are continuous.
So you can define (spoiler):
$f(x_1,x_2,x_3)= (+(Mcirc(p_1,p_1),Mcirc(p_1,p_3)), +(Mcirc(p_2,p_2),Mcirc(p_1,p_3)) ) (x_1,x_2,x_3)$
For g you also need the absolute value function $|cdot|$ to be continuous.
answered Jan 14 at 16:42
Felix B.Felix B.
739217
edited Jan 14 at 16:50
answered Jan 14 at 16:42
Felix B.Felix B.
739217
answered Jan 14 at 16:42
Felix B.Felix B.
739217
answered Jan 14 at 16:42
Felix B.Felix B.
739217
739217
add a comment |
add a comment |
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