Need clarification onto the intuition of why every continuous function on a compact set is uniformly continuous

Question

So you can have a function on (a,b) that is not uniformly continuous, but as soon as you ADD points to the domain it becomes uniformly continuous.. I've read through the technical arguments and can follow them, yet something seems rather peculiar still. Like, you're adding points, so it seems it should be harder to find a delta that works for everything, you know?

So it must be that there is something special about the points added. I'd appreciate clarity on this matter! Thanks a ton!

Answer 1

Note that if a function is uniformly continuous in $[a,b]$ then it is uniformly continuous in $(a,b)$ as well. So if a function is not uniformly continuous in an open interval $(a,b)$ it means you can't simply "add points" and extend this function continuously to the closed interval $[a,b]$. Look for example at $f(x)=\frac{1}{x}$ in $(0,1]$. It is not uniformly continuous and you can't simply define it the point $x=0$ to make it continuous. It is impossible.