# How strong continuity is needed for the values of derivative on rational numbers to contain all information about the derivative?

Consider these different notions of continuity, listed in order of decreasing strength:

• Continuously differentiable
• Lipschitz continuous
• Absolutely continuous
• Uniformly continuous
• Continuous

I was wondering how far up the list we have to go before it is ensured that a function $f:\mathbb R\to\mathbb R$ must be such that $f'(x_0)$ for an irrational $x_0$ can always be recovered from information about only the values of $f'$ for rational numbers, i.e., that $f'(x_0)=\lim_{x\to x_0}f'|_{\mathbb Q}(x)$ whenever the right-hand side of that equation is defined.

I know that uniform continuity is not sufficient: Minkowski's_question_mark_function is a counter-example (for all rational $x$, $f'(x)$ is $0$, so the RHS of the equation is also always $0$, but there are irrational $a$ such that the LHS is not). On the other hand, $f$ being continuously differentiable is obviously sufficient. But what about absolute and Lipschitz continuity?

• Absolute continuous functions are characterized as those function having weak-derivative. In particular a continuous function $f$ is absolutely continuous whenever it has a "weak derivative" $g$, i.e. a function such that $f(x)= f(0)+ \int_0^x g(t) \ \mathrm{d}t$. – Crostul Jan 23 '17 at 18:36
• @Crostul: But is the weak derivative guaranteed to have the property I'm after? – Casper Jan 23 '17 at 20:23

Lipschitz continuous is not enough. Enumerate rational points as $q_1,q_2,\dots.$ Let $U=\bigcup_{n=1}^\infty(q_n-2^{-n}, q_n+2^{-n})$. This is an open set of measure at most $1$ which contains all rational points. Let $$f(x)=\begin{cases} m([0,x]\setminus U)\,\quad & x>0\\ 0,\quad & x\le 0\end{cases}$$ where $m$ is the Lebesgue measure. This is a nonconstant $1$-Lipschitz function. Its derivative is equal to $0$ at all rational points, by construction.
By the way, $f'(x)=1$ for a.e. $x\in (0,\infty)\setminus U$, by the Lebesgue density theorem.