If I know that $f:(0,\infty)\rightarrow \Bbb R$ is uniformly continuous on the intervals $[a,\infty)$ and $(0,a]$, where $a$ is in $(0,\infty)$, how can I prove that it is uniformly continuous on $(0,\infty)$? I know the general definition of uniform continuity using epsilon-delta, but I am not sure how to apply it to the above.
Thanks
Edit: I meant Uniformly continuous on the first two intervals