Continuous function sends closed sets on closed sets

1. Aug 8, 2006

buddyholly9999

Let $$f: D \rightarrow \mathbb{R}$$ be continuous.

Is there an easier function that counterexamples;
if D is closed, then f(D) is closed
than D={2n pi + 1/n: n in N}, f(x)=sin(x) ?????

Plus, these counterexamples are very similar ...but are they correct?

If D is not closed, then f(D) is not closed.
CE: D = (0, 1) and f(x) = 5
If D is not compact, then f(D) is not compact.
CE: We use same CE as above
If D is infinite, then f(D) is infinite.
CE: D = all real numbers and f(x) = 5
If D is an interval, then f(D) is an interval
CE: Use same CE as first

Last edited: Aug 8, 2006
2. Aug 8, 2006

matt grime

Don't double post.

3. Aug 8, 2006

buddyholly9999

Couldn't figure out how to delete these mofo's.....so anything to add to my question....?