Originally Posted by
nice rose - true or false:
every compact set in R(p) is connected in R(p)
What is R(p)?
- find an example of a function is uncontinuous at x in it is domin but it has a limit at x
$\displaystyle f(x):= \left\{\begin{array}{cc}1&\,\mbox{if } x\neq 0\\0&\,\mbox{if } x=0\end{array}\right.$
- f: R--------> R continuous proof that f([2,3]) is compact set in R
Continuous functions map intervals to intervals, and by boundness of f it must be that $\displaystyle f([2,3])$ is a bounded interval. Now, just prove (using limits, say) that this interval is actually closed and there you are .
Thanks