Strong differentiability condition

[Office_Shredder]

The reason you exclude the value that you're taking the limit towards is because you often have a function f(x) which is not defined at x=a, but is continuous elsewhere, and you want to know if you can extend the function continuously to a value at x=a. You can write this in shorthand notation as whether the limit
\lim_{x\to a} f(x)
exists, but only if the definition of the limit does not permit plugging x=a into the formula for f(x). If it does then the limit trivially doesn't exist because the function's original domain didn't include that point. It would be easy enough to work around but new notation/definitions of other things would have to be constructed to maintain rigor

 

[pwsnafu]

Is there any situation, where we actually have a reason to forbid some points from inside the domain during limit?

When you sum a Fourier series (assuming it is convergent), the value at the end points (x=L and x=-L) are not necessarily equal to the limits at the endpoints.

Edit: Just to elaborate.

I'm talking about the case of one period, $f : [-L,L] \to \mathbb{R}$. Open neighborhoods of L are of form $(L-\epsilon, L]$. You need to remove L to take the limit.

Also, notice that the limit of L is always one-sided, so it always exists. (You don't have the situation where left is different to right and hence non-existing.)