Modifications of Thomae's Function and Differentiability

As we shall see later, Thomae’s function is not differentiable on the irrationals. In this note, we address whether there is a modification of Thomae’s function which is differentiable on a subset of the irrationals. In Section 2, we prove that Thomae’s function is not differentiable on the irrationals and define modified versions of Thomae’s function. In Section 3, we show that for each of our modifications there is a dense subset of irrationals on which, quite surprisingly, the function is not differentiable. Finally, in Section 4, we show that the measure of irrationality of a given number determines which modifications of Thomae’s function are differentiable at that number.