| 概要 | It has been more than a century since Teiji Takagi gave us the
function now commonly bearing his name. Yet this simple example
of a continuous but nowhere differentiable function keeps fascinating
and inspiring mathematicians as never before, not least because
of its wide variety of applications to such diverse areas as number
theory, combinatorics, and analysis. It has even been linked to
the Riemann hypothesis. The talk will begin with an overview of
these applications, to make it clear that the Takagi function is
much more than just 'a peculiar function' and that it is actually
a function worth studying in more detail. We will then see that
studying the Takagi function is also intrinsically rewarding, as
it possesses many deep and beautiful secrets that have only recently
started to come out into the light. These include the location of
infinite derivatives and its rich level set structure. In the last
part of the talk we will look at a natural generalization of the
Takagi function, and compare its properties with those of the
Takagi function itself. |