# $\epsilon - \delta$ definition of continuity

I'm trying to give an $$\epsilon - \delta$$ definition for a function $$f: \mathbb{R}_l \rightarrow \mathbb{R}$$ to be continuous, where $$\mathbb{R}_l$$ denote $$\mathbb{R}$$ with lower limit topology. Here is my attempt: $$f: \mathbb{R}_l \rightarrow \mathbb{R}$$ is continuous at $$p \in \mathbb{R}_l$$ if given $$\epsilon > 0$$, there exists $$\delta \geq 0$$ such that if $$\vert p - x \vert \leq \delta$$, then $$\vert f(p) - f(x) \vert < \epsilon$$. It this correct?

What you provided was the usual definition of continuity. It should be:$$(\forall\varepsilon>0)(\exists\delta>0):p\leqslant x