Let $f:(0,1) \to \mathbb{R}$ be a given function.
Definition: for any $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for all $x \in (0,1)$ and $|x-x_0| \leq \delta$, one has $|f(x) - f(x_0)| \leq \epsilon$ .
I think its equivalent to the definition of continuity because it doesn't matter if it's $\lt$ or $\leq$.