Definition (Vitali Cover)
Let and let be a collection of intervals.
We say is a Vitali cover of if for every and every , there exists
such that
Example
Take , . Then
is a Vitali cover of .
Proof: exercise.
Vitali Covering Lemma
Suppose with and is a Vitali cover of .
Then for any , there exists a finite disjoint collection
such that
Dini Derivatives
Suppose and . Define:
- Upper right derivative
- Lower right derivative
- Upper left derivative
- Lower left derivative
If all four limits are finite and equal, then is differentiable at .
Examples
- Dirichlet-type function on :
At , the Dini derivatives are infinite in the expected oscillatory sense
(exercise: compute all four explicitly).
- :
- : compute Dini derivatives at (exercise).
Basic inequalities
Always,
In general, relations like need not hold (example: Cantor function).
Exceptional Sets and Differentiability Set
Define subsets of :
Let
Proposition
Idea of proof
If , then the chain of inequalities between the four Dini derivatives at forces
all of them to be finite and equal. Hence is differentiable at .
Exercise:
Theorem (Lebesgue-Young)
If is monotone on , then is differentiable a.e. (with respect to Lebesgue measure).
Proof outline from class notes
Assume is increasing.
It suffices to show
For a fixed , for and small one has
Thus intervals of the form give a Vitali cover of .
Apply Vitali covering lemma to extract finite disjoint intervals and use monotonicity to estimate
which yields after letting parameters vary.
The same method handles , hence the theorem.