# Function that is defined for all reals and is continuous but not uniformly continuous

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?

• You can use $f(x) = x^2$. Dec 20, 2021 at 14:58
• 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. Dec 20, 2021 at 15:04
• There's some good commentary on the wiki page for uniform continuity 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). Dec 20, 2021 at 15:05
• Possibly of interest Dec 20, 2021 at 15:09

$$𝑓(𝑥)= x^2$$