  Given a real valued and time-inhomogeneous martingale diffusion X, we
investigate the properties of functions defined by the conditional expectation
f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitz
continuous then f(t,x) will also be monotonic or Lipschitz continuous in x. If
g is convex then f(t,x) will be convex in x and decreasing in t. We also define
the marginal support of a process and show that it almost surely contains the
paths of the process. Although f need not be jointly continuous, we show that
it will be continuous on the marginal support of X. We prove these results for
a generalization of diffusion processes that we call `almost-continuous
diffusions', and includes all continuous and strong Markov processes.
