  In this paper we look at the properties of limits of a sequence of real
valued time inhomogeneous diffusions. When convergence is only in the sense of
finite-dimensional distributions then the limit does not have to be a
diffusion. However, we show that as long as the drift terms satisfy a Lipschitz
condition and the limit is continuous in probability, then it will lie in a
class of processes that we refer to as almost-continuous diffusions. These
processes are strong Markov and satisfy an `almost-continuity' condition. We
also give a simple condition for the limit to be a continuous diffusion. These
results contrast with the multidimensional case where, as we show with an
example, a sequence of two dimensional martingale diffusions can converge to a
process that is both discontinuous and non-Markov.
