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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?

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What you provided was the usual definition of continuity. It should be:$$(\forall\varepsilon>0)(\exists\delta>0):p\leqslant x<p+\delta\implies\bigl\lvert f(x)-f(p)\bigr\rvert<\varepsilon.$$

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