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I'm trying to resolve what seems like an inconsistency between the epsilon-delta definition of continuity and the limit-based definition ($\lim_{x->c} f(x) = f(c)$). Assume $c$ is a cluster point. There are many questions about what happens when there are singularities, for example asking whether $\frac{1}{x}$ is continuous at $x=0$, and the answers indicate that continuity is relative to the function's domain (for example here and here.) However, what happens when the function is defined at that point? For example, take $f(x) = \frac{1}{x}, x\neq0$ but $f(x)=0$ at $x=0$. According to the epsilon-delta definition of continuity this function would be continuous, because at any point you can find an open ball of sufficiently small radius $\delta$ in the domain for the corresponding radius $\epsilon$ in the function's range (because we require that $\epsilon>0$, not $\epsilon\geq0$ in this definition). However, the limit is clearly not equal to the function value at 0, so by that definition the function would be discontinuous. Another example is the example of a removable discontinuity on Wikipedia. Is this just a question of being clear on which definition is used in a given context? I can imagine that in many contexts the distinction between these definitions could matter.

I've found many similar questions, but none that quite address my question (this one is the closest).

Vyas
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    The function is discontinuous at $x=0$ (as you might imagine). The $\epsilon-\delta$ definition of continuity requires that $|f(x)-f(0)|<\epsilon$ FOR ALL $x$ such that $|x|<\delta$, not just one particular $x$. – David M. Mar 19 '19 at 17:01
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    I was mixing up the definition of limits and continuity. The limit of a function at a point can exist by just considering the function on the complement of that point in the domain, but for continuity that limit does have to match the value at the point. Didn't click until I read your comment, thanks! – Vyas Mar 19 '19 at 17:07

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Copying my comment into an answer: I was mixing up the definition of limits and continuity. The limit of a function at a point can exist by just considering the function on the complement of that point in the domain, but for continuity that limit does have to match the value at the point. So my examples would indeed be discontinuous functions, but the limits of those functions at the point of discontinuity are well-defined.

Vyas
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