Continuity and Boundedness: Let f:R^n --> R^m be continuous

[wickeddude12]

Having proven this:
[attachment=0:26bsj90p]Proof.PNG[/attachment:26bsj90p]

How do you explain the counterexample below? Or rather, how is it not a counterexample at all?
[attachment=1:26bsj90p]Counterexample.PNG[/attachment:26bsj90p]

 

[pka]

Re: Continuity and Boundedness

Are you saying that \(\displaystyle \frac{1}{{x^2 }}:\mathbb{R}^1 \mapsto \mathbb{R}^1\)
is a continuous function?

 

[wickeddude12]

Re: Continuity and Boundedness

I think I see it now: so you are saying that f must be continuous on all of R before you take a bounded subset of R for the statement to hold?