Prove the following function isn't uniformly continuous

Asked Sep 21 '17 at 10:30

Active Sep 21 '17 at 10:43

Viewed 68 times

I need to prove the function

$f(x)=x\ln(x)$

is not uniformly continuous on the interval $(0,\infty)$ I have tried using some theorems about uniform continuity but non of which apply in this case.

Any help is much appreciated.

~Yuval

edited Sep 21 '17 at 10:43

asked Sep 21 '17 at 10:30

Yuval Gat

1,375

equicontinuous is not the word you want – Bernard W Sep 21 '17 at 10:35
@Bernard8 How so? that's precisely the question I need to solve just translated to English :P – Yuval Gat Sep 21 '17 at 10:36
equicontinuity is a property that can be held by a collection of functions (definitely more than one). Your question only refers to an individual function. – Bernard W Sep 21 '17 at 10:38
@Bernard8 In my book, equicontinuity of a single function is defined to be:
$f(x)$ is equicontinuous if and only if $\forall \varepsilon > 0, \exists \delta > 0, |x_1-x_2|<\delta\implies |f(x_1)-f(x_2)|<\varepsilon$
– Yuval Gat Sep 21 '17 at 10:39
Ah, that's uniform continuity – Bernard W Sep 21 '17 at 10:42
1

Ah thanks! I didn't know. Fixed. – Yuval Gat Sep 21 '17 at 10:43
See this. – David Mitra Sep 21 '17 at 11:13

Prove the following function isn't uniformly continuous

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