  In the recent article [A. Jentzen, B. Kuckuck, T. M\"uller-Gronbach, and L.
Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions
to every additive noise driven stochastic differential equation (SDE) which has
a drift coefficient function with at most polynomially growing first order
partial derivatives and which admits a Lyapunov-type condition (ensuring the
the existence of a unique solution to the SDE) depend in a logarithmically
H\"older continuous way on their initial values. One might then wonder whether
this result can be sharpened and whether in fact, SDEs from this class
necessarily have solutions which depend locally Lipschitz continuously on their
initial value. The key contribution of this article is to establish that this
is not the case. More precisely, we supply a family of examples of additive
noise driven SDEs which have smooth drift coefficient functions with at most
polynomially growing derivatives whose solutions do not depend on their initial
value in a locally Lipschitz continuous, nor even in a locally H\"older
continuous way.
