let f(x) be defined on [a,b]

Continuity

Def 1)

Lim f(x) requires neighborhood in (a,b) → 0.

Lim f(a) not defined so f(x) not continuous at a.

Def 2)

Lim f(x) requires neighborhood in [a,b] → 0.

If x ≠ a, same as Def 1). If x = a, definition of continuity differs from x ≠ a.

f(x) not continuous at a because you can’t change definition of continuity for x ≠a and x = a (in mid-stream).

Same applies for derivatives.

It becomes an issue if you require conditions at a boundary determine conditions in the interior.