Cantor type functions in non-integer bases

Cantor's ternary function is generalized to arbitrary base-change functions in non-integer bases. Some of them share the curious properties of Cantor's function, while others behave quite differently

A PROOF OF NEWTON\u27S POWER SUM FORMULAS

For a polynomial P(z) = α0 + α1z + ... + αnzn = αn (z – z1) (z – z2) ... (z – zn), the power sums Sm = ∑ n k=1 zm k, m = 1, 2, ... , can be calculated from the formulas ..

THE DIFFERENTIABILITY OF \u3ci\u3ea\u3csup\u3ex\u3c/sup\u3e\u3c/i\u3e

A from scratch proof of the differentiability of ax, a \u3e 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define ax as ex log a its differentiability and other properties following from that of the functions ex and log x. Unfortunately, the usual definitions of ex and log x involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of e, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is told that e is e1. He would have a much better feeling for the naturalness of e if it were defined as that number a for which (ax)\u27 = ax. The purpose of this note is to provide a direct and relatively simple way of getting at the differentiability of ax