Continuous function sends closed sets on closed sets

[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

 

[matt grime]

Don't double post.

 

[buddyholly9999]

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