Show that f is uniformly continuous on [0, +∞)

Question

So working on an exercise from my notes, I am given the conditions that $f$ is continuous on $[0, +∞)$ and uniformly continuous on $[a,+∞)$ for some $a > 0$.

How do I show that $f$ is uniformly continuous on $[0, +∞)$?

Answer 1

You know that it is uniformly continuous on $[0,a+1]$ by compactness of that set. Then for $[0,a+1]$ and $[a,\infty)$ there are $\delta_1, \delta_2$ such that $|x-x_0|<\delta_1$ and $x_0\in [0,a+1]$ or $|x-x_0|<\delta_2$ and $x\in [a,\infty)$ we have $|f(x)-f(x_0)|<\epsilon$ for any prescribed $\epsilon >0$. Let $\delta=\min\{\delta_1,\delta_2\}$ and you get uniform continuity on $[0,\infty)$.