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To grasp the concept of Uniform Continuity, I was looking at examples from previous questions of functions that are continuous on a defined region A, but not uniformly continuous.

But that begs the question about whether or not there exists a function that is defined for all x ∈ R, that is continuous, yet is not uniformly continuous? I couldn't find any such examples, does such a function exist?

Bob Smith
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    You can use $f(x) = x^2$. – Elchanan Solomon Dec 20 '21 at 14:58
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    Thanks, that's exactly what I was looking for. I wonder why every example I look at keeps on using $1/x$ defined on (0,1), I think $x^2$ is a much better example. – Bob Smith Dec 20 '21 at 15:04
  • There's some good commentary on the wiki page for uniform continuity https://en.wikipedia.org/wiki/Uniform_continuity:

    "Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as $$f(x) = \frac{1}{x}$$ on (0,1), or if their slopes become unbounded on an infinite domain, such as $$f(x) = x^2$$ on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).

     – Brian Lai Dec 20 '21 at 15:05
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    Possibly of interest – Andrew D. Hwang Dec 20 '21 at 15:09

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$()= x^2$

Thanks to Elchanan Solomon.

Bob Smith
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