On the existence of a derivative continuous on a dense G δ.

The main object of this paper is to prove the following theorem: Theorem . Let S ⊆ [a,b ] Then there exists f : [a,b ] → R such that f is differentiable on [a,b ], f ′ is bounded on [a,b ] and C = { x ∈ [a,b ], f ′ is continuous at x} = S if and only if S is a G δ set dense in [a,b ]. In section 5, consequences of the above theorem explore how badly discontinuous a derivative can be. Among these it will be shown that there exists a function f : [a,b ] → R such that f ′ is a Dirichlet function, and it will be shown that there exists no function f : [a,b ] → R such that D f′ = [ a , b ] - C f′ is an interval, and an example of a 'worst case' scenario is provided. The arguments stay within the realm of elementary classical analysis and are thus accessible to students who have encountered a first proof course in the subject. [ABSTRACT FROM AUTHOR]