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.

  • $\begingroup$ yes , that I know , but what to do next , how to choose the $\delta $ depending only on $\epsilon >0 $ ? $\endgroup$
    – Souvik Dey
    Aug 16 '14 at 13:29
  • $\begingroup$ 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 $\endgroup$ Aug 16 '14 at 13:34
  • $\begingroup$ You have a $\delta_1$ in $[0,a+1]$ and a $\delta_2$ in $[a,+\infty)$. Take $\delta=\min(\delta_1,\delta_2)$. $\endgroup$ Aug 16 '14 at 13:35

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