# continuous on $[0,\infty)$ and uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , to show uniform continuity on $[0, \infty)$

Let $f:[0, \infty) \to \mathbb R$ be a continuous function which is uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , then how to show that $f:[0, \infty) \to \mathbb R$ is uniformly continuous ? Please help

Hint: $f$ is uniformly continuous in $[0,a+1]$ because is a compact set.
• yes , that I know , but what to do next , how to choose the $\delta$ depending only on $\epsilon >0$ ? Aug 16 '14 at 13:29
• Find $\delta_1$ for $[a,\infty)$ and $\delta_2$ for $[0,a+1]$ and let $\delta=\min(\delta_1,\delta_2,\frac{1}{2})$... @SouvikDey Aug 16 '14 at 13:34
• You have a $\delta_1$ in $[0,a+1]$ and a $\delta_2$ in $[a,+\infty)$. Take $\delta=\min(\delta_1,\delta_2)$. Aug 16 '14 at 13:35