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Show that a mapping is continuous

  • #1
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Homework Statement


Show that the mapping f carrying each point [itex](x_{1},x_{2},...,x_{n+1})[/itex] of [itex]E^{n+1}-0[/itex] onto the point [itex](\frac{x_{1}}{|x|^{2}},...,\frac{x_{n+1}}{|x|^{2}})[/itex] is continuous.

2. Continuity theorems I am given.
A transformation f:S->T is continuous provided that if p is a limit point of a subset X of S then f(p) is a limit point or a point of f(X).

Let f:S->T be a transformation of space S into space T. A necessary and sufficient condition that f be continuous is that if O is an open subset of T, then its inverse image [itex]f^{-1}(O)[/itex] is open in S.

A necessary an sufficient condition that the transformation f:S->T be continuous is that if x is a point of S, and V is an open subset of T containing f(x) then there is an open set U in S containing x and such that f(U) lies in V.

The Attempt at a Solution


I was thinking to prove this I would have to find an open set in the range of this function and show that its inverse image is also open in the domain but I am not sure how I would go about doing that.

Any help would be appreciated.
 
Last edited:

Answers and Replies

  • #2
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390
What is the definition of continuity you work with?
 
  • #3
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I have three. They are

A transformation f:S->T is continuous provided that if p is a limit point of a subset X of S then f(p) is a limit point or a point of f(X).

Let f:S->T be a transformation of space S into space T. A necessary and sufficient condition that f be continuous is that if O is an open subset of T, then its inverse image f^(-1)(O) is open in S.

A necessary an sufficient condition that the transformation f:S->T be continuous is that if x is a point of S, and V is an open subset of T containing f(x) then there is an open set U in S containing x and such that f(U) lies in V.

I updated the original post to reflect this.
 
  • #4
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390
I would use the first one. Given a sequence ## \mathbf{x}^{(n)} \rightarrow \mathbf{p} ##, prove that ## \frac {\mathbf{x}^{(n)}} {(x^{(n)})^2} \rightarrow \frac {\mathbf{p}} {p^2} ##.
 
  • #5
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I would use the first one. Given a sequence ## \mathbf{x}^{(n)} \rightarrow \mathbf{p} ##, prove that ## \frac {\mathbf{x}^{(n)}} {(x^{(n)})^2} \rightarrow \frac {\mathbf{p}} {p^2} ##.
I don't really understand. Can you elaborate a little bit? I don't know how I would show that a point [itex]x=(x_{1},x_{2},...,x_{n+1})[/itex] is a limit point.
 
  • #6
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390
You don't have to prove ## \mathbf{x}^{(n)} \rightarrow \mathbf{p} ##, this is an assumption. Given that assumption, prove ## \frac {\mathbf{x}^{(n)}} {(x^{(n)})^2} \rightarrow \frac {\mathbf{p}} {p^2} ##.
 

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