system
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human
[Limit definition] Why can't |f(x)-L| ≤ ε ?

I'm curious. I'm just seeing no problem with (0<|x-a|≤ 𝛿 => |f(x)-L|≤ ε). Why couldn't this be the definition instead?

Also, why isn't ( 0<|f(x)-L|< ε) a part of the definition?

Any help is really appreciated!
gpt
I’ll answer your second question first. The reason we take the punctured delta neighborhood of x is because if we allowed x-a to be 0 (ie if x=a) the epsilon condition would require the function to be both defined and continuous at a, and for limits we don’t want to be so restrictive; we only care what happens arbitrarily close to a. As for why we don’t puncture the epsilon neighborhood of f(x), if we did require 0<|f(x)-L|<ε, then we would be imposing the condition that f cannot take the value of L near x, and this would be unreasonable (constant functions, for example, would no longer be continuous). 

For your first question, we absolutely could use the non strict inequalities. This has to do with the (standard/Euclidean) topology on the real numbers. Using the non strict inequalities would result in closed intervals, and we can form the definition of a topology equally well in terms of closed sets. The reason we use the open set formulation (with the strict inequality <) is because in the real numbers, open sets are more well-behaved and are easier to work with than closed sets (every open set is a countable union of open intervals whilst closed sets can get more pathological; see the Cantor set for example). 

Here’s something for you to think about: would these definitions work equally well if we had strict inequality for the delta term but non strict inequality for the epsilon term (or vice versa)? Why or why not?