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".