  We introduce and compare three differential correlation functions for the
study of transverse momentum correlation in $p+p$ and $A+A$ collisions. These
consist of {\it inclusive}, {\it event-wise} and a differential version of the
correlation measure $\tilde C$ introduced by Gavin \cite{Gavin} for
experimental study of the viscosity per unit entropy of the matter produced in
$A+A$ collisions. We study the quantitative difference between the three
observables on the basis of PYTHIA simulations of $p+p$ collisions and $A+A$
collisions consisting of an arbitrary superposition of $p+p$ collision events
at $\sqrt{s} = $200 GeV. We observe that {\it inclusive} and {\it event-wise}
correlation functions are remarkably identical to each other where as the
observable $\tilde C$ differs from the two. We study the robustness and
efficiency dependencies of these observables based on truncated Taylor
expansions in efficiency in $p+p$ collisions and on the basis of Monte Carlo
simulation using an adhoc detector efficiency parameterization. We find that
all the three observables are essentially independent of detector efficiency.
We additionally study the scaling of the correlation measures and find all the
observables exhibit an approximate $1/N$ dependence of the number of
participants ({\it N}) in $A+A$ collisions. Finally, we study the impact of
flow-like anisotropy on the {\it inclusive} correlation function and find flow
imparts azimuthal modulations similar to those observed with two-particle
densities.
