Discontinuity set of derivative

Let $f$ be a differentiable function and let $J$ and $D$ denote the set of continuity and discontinuity of $f'$. Then $J$ and $D$ can be characterized as done here.

Also $J$ is a $G_{\delta}$ dense set and hence has positive measure (?)

Now Volterra function can be used to construct a derivative which is discontinuous on a nowhere dense set with both zero and positive measure by choosing appropriate Cantor sets.

Hence $D$ can be a nowhere dense set with positive measure. This is where I am struck.

Now if $D$ has full measure, then $J$ has zero measure which is a contradiction that $J$ is a $G_{\delta} $ dense set?.

Solution 1:

“Also $J$ is a $G_{\delta}$ dense set and hence has positive measure”

This is not true, and proofs of a stronger result are given in intuition of decomposition of $\mathbb R$ into disjoint union of first category and null set.

To help in understanding how these properties relate to each other, let’s consider the following three properties for subsets of $\mathbb R$ --- countable, meager, and zero Lebesgue measure. Each of these properties conveys (i) a notion of “small”, (ii) a notion of “large”, and (iii) a notion of “maximally large”:

small notions: $\;$ countable,$\;$ meager,$\;$ zero Lebesgue measure

large notions:$\;$ uncountable,$\;$ non-meager,$\;$ positive Lebesgue measure

maximally large notions:$\;$ co-countable,$\;$ co-meager,$\;$ full Lebesgue measure

The “maximally large” notions are those for which the set is so large that what’s left over is small (i.e. the complement of the set is small).

The three following observations should help in beginning to get a feel for how these notions relate to each other.

(1) Each of meager and zero Lebesgue measure is a weaker notion of being small than countable.

(2) Each of non-meager and positive Lebesgue measure is a stronger notion of being large than uncountable.

(3) Each of co-meager and full Lebesgue measure is a weaker notion of maximally large than co-countable.

The notions “countable” and “meager” (= first Baire category) can be defined via countable unions of smaller building blocks --- every countable set is a countable union of finite sets (equivalently, a countable union of singleton sets) and every meager set is a countable union of nowhere dense sets. There is no natural smaller building block notion for the zero Lebesgue measure sets.

Each subset of a small set is also a small set (of the same type) and each countable union of small sets (all of the same type) is small set (of the same type). Also, for each type of small set, $\mathbb R$ is not a small set. Putting the last two sentences together gives the result that for each type of small set, each countable union of that type of small set will not be all of $\mathbb {R}.$ In fact, each countable union of that type of small set is not remotely close to being all of $\mathbb {R},$ since each countable union will be a small set, and hence what remains in $\mathbb R$ is a maximally large set (of the same type).

As indicated above, the notions related to “countable” are strictly comparable to the notions related to “meager” and “zero Lebesgue measure”. However, there is no comparability between the notions related to “meager” and “zero Lebesgue measure”. Indeed, it is possible for a set to be meager and not have zero Lebesgue measure (e.g. a Cantor set of positive measure; note this example shows that even a building block for the meager sets can fail to have zero Lebesgue measure), and it is possible for a set to have zero Lebesgue measure and not be meager (see intuition of decomposition of $\mathbb R$ into disjoint union of first category and null set).

In fact, it is possible for a set to be meager and to have full measure (i.e. small for Baire category and maximally large Lebesgue measure), and it is possible for a set to have zero Lebesgue measure and to be co-meager (i.e. small for Lebesgue measure and maximally large for Baire category). Indeed, the stack exchange thread cited above shows that $\mathbb R$ can be written as the union of two small sets (of different types), that is, $\mathbb {R} = A \cup B$ where $A$ is meager and $B$ has zero Lebesgue measure (the term orthogonal is sometimes used for a pair of smallness notions that have this property), and both statements in the previous sentence follow from this --- $A$ is small for Baire category and maximally large for Lebesgue measure, and $B$ is small for Lebesgue measure and maximally large for Baire category.