If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded? If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?
($a$ and $b$ being finite numbers).
I tried proving and disproving it. Couldn't find an example for a non-bounded image. 
Is there any basic proof or counter example for any of the cases?
Thanks a million!

Hint: Prove first that a uniformly continuous function on an open interval can be extended to a continuous function on the closure of the interval. 
