The Invisible Realm coming from Limits and Infinities

There are "too many" things that defy our direct observation and intuition, especially those objects coming from limit or infinite sum, for example

1. the function which is continuous everywhere but isn't differentiable (infinite function sum)

2. the Vitali set (collection of infinite element)

3. the Cantor function (infinite parts)

But we know that they indeed exist and we can deal with some instances due to the definition and almost everywhere "monotone".

And many things change when we consider limit, for example,

1. the sequence of rational number can approach a irrational number.

2. the union of infinite closed sets can be a open set and the intersection of infinite open sets can be closed.