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.