Take a function $f: \mathbb{R} \mapsto \mathbb{R}$. For any given $\alpha > 0$, I would like to be able to find a $\beta > 0 $ such that $$ \forall x, x' \in \mathbb{R} : \ |x - x'| \leq \beta \implies |f(x) - f(x')| \leq \alpha.$$ If I am not wrong, this is an implication of uniform convergence.
Now I would like to modify this. Fix a number $r \in \mathbb{R}$ arbitrarily. For any given $\alpha > 0$, we would like to be able to find a $\beta>0$ such that $$ \forall x, x' \in [-r, r] : \ |x - x'| \leq \beta \implies |f(x) - f(x')| \leq \alpha.$$
So I would like to claim something similar to the uniform convergence but on a bounded interval. I think the latter one is less restrictive as, e.g., $f(x) = x^2$ satisfies the latter property but it is not uniformly continuous.
My question is: do we have a name for this version of uniform convergence?
Yes. It's called continuous. Any function that's continuous on a compact subset of $\Bbb R^n$ is necessarily uniformly continuous on that subset.