Consider f(x) = 1/(1+x^2) as a function from R to R with the usual metric. Are the following statements true or false (give proofs or counterexamples):

1. f(A) is open for all open sets A in R

2. f(A) is open for all closed sets A in R

thanks!

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- April 10th 2010, 10:32 AMswallenbergopen? closed? both?
Consider f(x) = 1/(1+x^2) as a function from R to R with the usual metric. Are the following statements true or false (give proofs or counterexamples):

1. f(A) is open for all open sets A in R

2. f(A) is open for all closed sets A in R

thanks! - April 10th 2010, 10:40 AMmaddas
1. Let A=R. Then A is open and f(A) = (0,1].

2 is obviously false (take A to be a singleton). Did you mean f(A) is*closed*for closed sets A? If so, use the same example as (1). - April 10th 2010, 10:47 AMswallenberg
- April 10th 2010, 10:48 AMswallenberg
- April 10th 2010, 10:49 AMmaddas
Is (0,1] open?

- April 10th 2010, 10:53 AMswallenberg
- April 10th 2010, 10:56 AMswallenberg