The discussion revolves around the concept of continuity and limits in the context of a function f(x). Participants explore the definitions and implications of limits, particularly focusing on the behavior of functions at specific points and the conditions under which limits can be evaluated. The scope includes theoretical aspects of limits, mathematical reasoning, and clarifications on continuity.
Participants exhibit disagreement regarding the evaluation of limits, particularly in the context of the piecewise function. There is no consensus on the correct limit value, as some argue for 1 while others suggest it could be 2. The discussion remains unresolved with competing views on the definitions and implications of limits and continuity.
Participants express uncertainty about the definitions and applications of δ and ε, as well as the implications of continuity for different types of functions. There are unresolved questions about the behavior of limits in piecewise functions and the conditions under which limits can be evaluated.
The distance form f(x) to L, f(x)- L (not f(x) alone), can be larger than \epsilon for some x, just not for x "sufficiently close" to a. Once x is within some distance of a, |f(x)- L| cannot exceed \epsilon. And that "distance" is, of course, \delta.
Close but not exactly. The value f(x) "reaches" when x reaches a is f(a). The question is what happens when x is close to a but not equal to a.