3
$\begingroup$

Show that the following function is uniformly continuous on $(-1,1)$

$f(x) = \begin{cases} {x \sin \frac{1} {x}}, & \text{ } x\in(-1,0)\cup(0,1) \\ 0, & \text{ }x = 0. \end{cases} $

We cannot use the theorem that a continuous function on a compact set K is continuous on K, because we don't have a compact set. I was told the following hint: "if a function is uniformly continuous on a set then it is also uniformly continuous on any subset of this set". I don't know exactly what to do with this information, can you help me ? :)

I know the definition of uniform continuity, I (should) know what open, closed, compact sets are.

1 Answers 1

6

Hint: Show that you can extend the definition to $[-1,1]$ and that it is continuous on the closed interval. Then use the theorem about uniform continuity.

  • 1
    @Hempo: Yes, exactly.2012-10-20