  The partition functions of free bosons as well as fermions on $AdS_2$ are not
smooth as a function of their masses. For free bosons, the partition function
on $AdS_2$ is not smooth when the mass saturates the Breitenlohner-Freedman
bound. We show that the expectation value of the scalar bilinear on $AdS_2$
exhibits a kink at the BF bound and the change in slope of the expectation
value with respect to the mass is proportional to the inverse radius of
$AdS_2$. For free fermions, when the mass vanishes the partition function
exhibits a kink. We show that expectation value of the fermion bilinear is
discontinuous and the jump in the expectation value is proportional to the
inverse radius of $AdS_2$. We then show the supersymmetric actions of the
chiral multiplet on $AdS_2\times S^1$ and the hypermultiplet on $AdS_2\times
S^2$ demonstrate these features. The supersymmetric backgrounds are such that
as the ratio of the radius of $AdS_2$ to $S^1$ or $S^2$ is dialled, the
partition functions as well as expectation of bilinears are not smooth for each
Kaluza-Klein mode on $S^1$ or $S^2$. Our observation is relevant for evaluating
one-loop partition function in the near horizon geometry of extremal black
holes.
