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!

If, for instance, $\limsup\limits_{x\rightarrow a^+} f(x)=\infty$, then given any $\delta>0$, one could find $x$ and $y$ with $a<x<\delta$ and  $a<y<\delta$ such that $|f(x)-f(y)|>1$.  Could $f$ then be uniformly continuous on $(a,b)$?
Note if $f$ is continuous on $(a,b)$ and unbounded, then either $f$ is "unbounded at $a$" or "unbounded at $b$" (aside from the above, there are three other cases to consider).  
