- #1

- 274

- 1

## 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.

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.

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: