In 1909, Hardy gave an example of a transcendental entire function, f,
with the property that the set of points where f achieves its maximum
modulus, M(f), has infinitely many discontinuities. This is one of only
two known examples of such an entire function. Recently, we have
significantly generalised these examples. In particular, we have shown
that, given an increasing sequence of positive real numbers, tending to
infinity, there is a transcendental entire function, f, such that M(f)
has discontinuities with moduli at all these values. We also show that
the transcendental entire function lies in the much studied Eremenko-Lyubich
class. This is joint work with Leticia Pardo Simón.