I had nothing better to do than to propose a pathological function.

The function cos (x) has an infinite number of zeros that are constantly spaced.

cos (x ^ 2) also has an infinity of zeros, but the spacing between them decreases as we go on to infinity.

cos (x ^ -2) maps this infinity of zeros into a finite interval, namely [-1,1].

I understand that my quest for a pathological function might stop here, but I felt that its zero limit did not exist was more intuitive, as it contained an infinite number of waves in any interval around x = 0 .

So anyway, to generate a flat line with an infinite number of holes over a finite interval around zero, I simply split cos (x ^ -2) by itself to get cos (x ^ -2 ) / cos (x ^ -2)

Clearly

$$ {cos (x ^ {- 2}) over {cos (x ^ {- 2})}} = 1; x neq sqrt[scriptstyle-2]{{ pi over2} +2 pi k} enspace or enspace sqrt[scriptstyle-2]{{3 pi over2} +2 pi k}, enspace k in mathbb {N} $$

I asked my classmates about this term, and they thought about it a little, even when I warned them of his strange behavior in the neighborhood. They were all convinced that it was 1, since the function is 1 "almost everywhere" near x = 0.

I'll be honest, I'm not even sure it exists. This pathology prevents to define f for any interval around 0, which, in my opinion, could exclude the existence of a limit because you lose some possibilities to find an inner delta for each epsilon in an interval , since an infinity of epsilons is missing. .

I've watched the definition of a line-by-line boundary of wikipedia (ε, δ) (I've never followed a real course of analysis).

Let $ { displaystyle f} $ to be a real valued function defined on a subset $ { displaystyle D} $ real numbers. Let $ { displaystyle c} $ to be a limit point of $ { displaystyle D} $ and let $ { displaystyle L} $ to be a real number.

So, f is defined on a subset of real numbers, although an odd subset is. When I talk about D, it can be the interval [-1,1] or any subinterval including x = 0, as they are all annoying.

I have the feeling that the problem is the condition that c, here 0, must be a limit point of subset D of $ mathbb {R} $. To be honest, I only have an intuitive understanding of what is a limit point. I know that the purposes of an open interval are its endpoints, for example, but I have no idea what that is in general. For a normal space, I know that you can simply show that a point is a limit point if it is the limit of a sequence of points of the considered subset. So, to go back to this example, I think you can create any other sequence close to zero as long as you avoid the gaps.

So, get right to the point …

$$ { displaystyle lim _ {x to c} f (x) = L iff ( forall varepsilon> 0, , exists delta> 0, , forall x in D, , 0 <| xc | < delta Rightarrow | f (x) -L | < varepsilon)} $$

I will be honest, I do not quite understand this definition. In fact, I have never done any epsilon delta proof before.

If you could answer some of my reasoning, it would be great, but any hermetic proof of the state of the limit is appreciated.