Understanding the application of the Baire category theorem to show the continuous function $f:[0,1] \to \mathbb{R}$ is nowhere differentiable

Question

The Baire category theorem says that any compact Hausdorff space or a complete metric space is a Baire space.

A Baire space is when you take the union of a countable collection of close sets in a space $X$, each of whose interior is non empty in $X$, also has an empty interior is $X$.

I am trying to use this theorem to prove the continuous function $f:[0,1] \to \mathbb{R}$ is nowhere differentiable.

I understand how to do the first two steps which involve defining your sets and showing they are closed.

But I do not understand the general argument when trying to show your set is nowhere dense.

How do we do this?

Answer 1

The Baire Category Theorem can be used to proof the following theorem by Banach:

Let $D_+$ be the collection of all $f \in C[0,1]$ for which there is a point $x_f \in [0,1)$ at which $f$ has a finite right-hand derivative. Then $D_+$ is of the first category in $C[0,1]$.

It is from this statement that one can deduce that "a "typical" $f \in C[0,1]$ is not differentiable anywhere". The above statement is Theorem 1.5.5 in Megginson's An Introduction to Banach Space Theory, and I refer to this book for the proof as it is quite lengthy.