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Problem

For $p \in \mathbb{R}^1 $, let $f(p) = (2p+1,p^2)$.

a) Prove that $f:\mathbb{R}^1 \to \mathbb{R}^2 $ is uniformly continuous on the closed interval $[0,2]$.

b) What is the largest interval in which the given function is uniformly continuous?

$Attempt$

a) $4+(p_1+p_2)^2 \leq 4+(|p_1|+|p_2|)^2 \leq 4+(2+2)^2= 20$

$d(f(p_1),f(p_2))= (p_1-p_2)\sqrt(4+(p_1+p_2)^2) \leq \delta *\sqrt(20)= \epsilon$.

Hence f is uniformly continuous on [0,2].

Doubt

How to approach part (b) of the question?

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Any continuous function on a closed interval is uniformly continuous. So there is no largest interval on which $f$ is uniformly continuous. [It is not uniformly continuous on $\mathbb R$].

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