In Understanding Analysis by Stephen Abbott, I have understood that Baire's Category Theorem applied on C[0,1] tells us that there are much more nowhere-differentiable continuous functions than those with derivative at on one point.
And this implies that most of continuous functions cannot be written as a countable union of nowhere-dense sets, and that continuous functions that are differentiable at least one point can be written as a countable union of nowhere-dense sets.
However, I am not sure if I understand what it means for a function to be written as a countable union of nowhere-dense sets.
Is there a visual, intuitive way to see how can a function be written or not written as a union of sets?
And why can a continuous function that has derivative at least one point be expressed as a countable union of nowhere-dense sets?
Quoting Abbott,
"Now, in this space we will se that the collection of continuous functions that are differentiable at even one point can be written as the countable union of nowhere-dense sets." (97)