Continuous function vs Uniformly continuous function

Question

Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of $\delta$ depends only on $\varepsilon$ and not on the point in the domain.'' But in both definitions there's only $\exists \delta >0$

Answer 1

Generally for continuity when we write $\delta,$we mean that $\delta=\delta(\epsilon,x_0),x_0\in D$. Similarly for uniform continuity we mean $\delta=\delta(\epsilon)$. This notation is consistent. It is taken for granted that we understand the situation in which $\delta$ is referred.

Now for the example :
$f(x)=x^2$ in $\mathbb{R}$ is continuous but not uniformly continuous. But $f(x)=x$ is uniformly continuous in $\mathbb{R}$.

Note that : Uniform continuity $\implies$ continuity, but converse is not true.

Answer 2

Continuity is a condition on the function for given single points in the domain, while uniform continuity is a condition on the function for given pairs of points in the domain. Often one is sloppy with the quantifiers, but conditions of uniform continuity should start with two $\forall x_1 \forall x_2$ and then the $\exists \delta>0$. The condition for continuity only got one $\forall x$ before the $\exists\delta>0$ and one after.