## If $f$ is uniformly continuous on $[a,\infty)$,and $f’$ has definition on $[a,\infty)$,can one deduce that $f’$ is bounded?

If $$f$$ is uniformly continuous on $$[a,\infty)$$,and $$f’$$ has definition on $$[a,\infty)$$,can one deduce that $$f’$$ is bounded on $$[a,\infty)$$?

I know some functions like $$\sqrt{x}$$ which has definition on the open set $$(0,\infty)$$ , but which derivative is not bounded, so I wish the boundedness will be hold on a closed set $$[a,\infty)$$