  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,
$\mathcal{M}(f)$, has infinitely many discontinuities. This is one of only two
known examples of such a function.
  In this paper we significantly generalise these examples. In particular, we
show that, given an increasing sequence of positive real numbers, tending to
infinity, there is a transcendental entire function, $f$, such that
$\mathcal{M}(f)$ has discontinuities with moduli at all these values. We also
show that the transcendental entire function lies in the much studied
Eremenko-Lyubich class. Finally, we show that, with an additional hypothesis on
the sequence, we can ensure that $f$ has finite order.

Complex Variables