Example of continuous function that isn't uniformly continuous and isn't 1/x

Question

I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and $H$ is not an open interval, what is an example a function that satisfies continuity but not uniform continuity ($H$ and $f$, please)?

Answer 1

I think you have in mind that the open interval is also bounded when you say "I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities".

Let $\mathbb{R}=(-\infty,\infty)$ (you can view $\mathbb{R}$ as an interval which hopefully explains my previous comment) then $f(x)=x^{2}$ is not uniformly continuous. (You can negate the definition of uniform continuity and show that the statement you obtain holds true. It will take some doing, but the basic idea is that as long as you can pick your numbers from the real line, you can pick two numbers really close together such that their squares are far apart. For example $10^{10}+\frac{1}{10^{10}}$ and $10^{10}$ are really close together but their squares are $2$ apart.)

Answer 2

$H$ be the half open interval $(0,1]$ and $f(x)=\sin\left(\frac{1}{x}\right)$.