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Theorem 3.39 of An Introduction to Analysis by W. R. Wade is:

Suppose that $I$ is a closed, bounded interval. If $f : I \to \mathbf{R}$ is continuous on $I$, then $f$ is uniformly continuous on $I$.

Is there such a thing as a closed, unbounded interval?

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Yes: all intervals of the forms $[a,\to)$ and $(\leftarrow,a]$ are closed and unbounded, and so is $\Bbb R$. These are all of the closed, unbounded intervals. – Brian M. Scott Oct 7 '12 at 11:47
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Yes, for example the interval $ [0, \infty)$ is clearly unbounded, and is closed in $\mathbb{R}$ because its complement is $(-\infty, 0)$ is open.

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