What does this mathematical sentence mean? "Let $f(x)$ be a function defined on an interval that contains $x=a$, except possibly at $x=a$."

Question

What does this mathematical sentence mean?

"Let $f(x)$be a function defined on an interval that contains $x=a$, except possibly at $x=a$."

How can a function contain $x=a$ "except possibly at $x=a$"? That is so illogical to me.

This is an excerpt out of the definition for $\lim$.

I know there are questions here that have discussed this already, however I think my question is a bit different as it focuses more on the definition as to what it says in relation to limits. My main question, again, is:

How can a function contain $x=a$ "except possibly at $x=a$"?

Answer 1

It means the same as "let $I$ be an interval containing $a$, and let $f$ be a function to $\mathbb R$ from either $I$ or $I\smallsetminus\{a\}$". Nothing specifies whether $I$ has to be open, closed, or finite, so it should allow any of these possibilities. In particular $I$ could be a closed interval with $a$ as an endpoint.

Answer 2

It means $f(x)$ is defined everywhere in the interval, except that it may or it may not be defined at $x=a.$