Call **a computable function** a total function $\mathbb{R} \to \mathbb{R}$, for which there exists a Turing machine outputting arbitrary close approximation to $f(x)$ given arbitrary close approximation to $x$.


1. Obviously not every computable function is differentiable (for example, absolute value).
For arbitrary continuous functions, the set of points of differentiability is $\Pi\_{3}^0$.
Can this be improved for computable functions?
2. Suppose $f$ is computable and continously differentiable everywhere. Must $f'$ be computable?



Differentiability of computable functions
John Myhill gave an example of *a recursive function defined on a compact interval and having a continuous derivative that is not recursive* [Michigan Math. J. 18 (1971), 97-98, [MR0280373](http://www.ams.org/mathscinet-getitem?mr=280373)]. However, Pour-El and Richards have shown that if a recursive function defined on a compact interval has a continuous second derivative, then it has a recursive first derivative [*Computability and noncomputability in classical analysis*, TAMS 275 (1983), 539-560, [MR0682717](http://www.ams.org/mathscinet-getitem?mr=682717)].


