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Pick out the true statement(s).

(a) If $f :(−1, 1)\to\Bbb R$ is bounded and continuous, it is uniformly continuous.

(b) If $f : S^1\to\Bbb R$ is continuous, it is uniformly continuous.

(c) If $(X, d)$ is a metric space and $A\subseteq X$, then the function $f(x) = d(x,A)$ defined by $$d(x,A) = \inf\{d(x, y) : y \in A\}$$ is uniformly continuous.

i think a is false as domain is not closed and b is true as domain is closed and bounded. no idea about c

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For (a), if it is false (which it is) one should produce an example. Hint: Lots of wiggle. –  André Nicolas Sep 6 '12 at 4:27
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You really should start learning how to use MathJax and format your posts clearly; there are lots of links and suggestions here to get you started. –  Brian M. Scott Sep 6 '12 at 4:30

1 Answer 1

up vote 0 down vote accepted

(c) is true. Just think about what $x \mapsto d(x, A)$ means geometrically. Moving $x$ by $k$ units cannot change $d(x, A)$ by more than $k$ units.

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