1. Some years ago Collingwood and Valiront proposed the problem of whether there could exist an integral function whose minimum modulus on every circle Iz I =r is bounded, but possessing no asymptotic paths. By an asymptotic path we mean a continuous path tending to infinity along which the value of the function tends to a limit. In this paper I show how to construct such a function. It is obtained by considering the well known Weierstrassian non-differentiable function