Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only such distances seems to be the Wasserstein metric, a.k.a. the earth mover's distance, and Kolmogorov-Smirnov in the one-dimensional case, but none of the others (e.g. f-divergence) seems to work. Have I missed some? I find it surprising that no such f-divergences exist.
TL;DR Look for metrics which metricize weak convergence.
After the comments above, I think I can answer your question to a reasonable degree. I am not a practicing statistian, so I can't comment too authoritatively on "usefulness". However, I am a computability theorist who specializes in measure theory, so I have a good sense of what is useful in theory (but maybe not in practice).
One way to think of a metric is to think of the topology it corresponds to. In my experience the weak topology (or the weak-$*$ topology in functional analysis terminology) is by far the correct topology to talk about measures/distributions. The other topologies are too strong. (I had originally misunderstood your question as wanting a strong topology---which separates more stuff. Instead, you wanted a weak topology---which puts things close together if they are close in a practical sense.)
First of all, the weak topology is good because the finite discrete measures are dense in this topology. Indeed, as @usul mentioned, you want the distributions of finitely many independent samples (the empirical distribution?) to converge to the true distribution. This happens almost-surely in the weak topology. (There is probably a result that says that no stronger topology works, but I don't know of a reference off hand.)
Now, the Wasserstein Metric isn't exactly a metric for the weak topology. It is a metric for weak convergence and convergence in the $p$th moment. In particular it only works if both the distributions have finite p-th moments. For example, if $\mu$ does not have $p$th moments, then it will have infinite distance to any finite discrete measure. Now, if you are only considering measures on a bounded metric space (and you can always bound a metric $d$ by just using $\rho(x,y) = d(x,y) \wedge 1$ or $\rho(x,y) = \frac{d(x,y)}{d(x,y) + 1}$) then the Wasserstein metric is a metric for weak convergence.
Now, (contrary what I said in the comments) the total variation distance separates things too much. Consider the uniform measure on $[0,1]$ and the uniform measure on $\{0,\ldots, k2^{-i}, \ldots, 1\}$. This has distance $1$ in the total variation metric, making it completely impractical for approximating a continuous measure with a discrete one. Moreover, it is even worse, the total variation metric is not separable, that is there is no countable dense set of measures $\mu_i$ in this metic. Therefore, there are measures that are not approximable in any practical way in this metric. (By practical, I mean you have some countable list of approximations to choose from.)
Now, what metrics topology weak convergence? The Levy-Prokhorov metric (any complete separable metric space). The Levy metric ($\mathbb{R}$). The Wasserstein metric (a bounded metric space). If your distribution is on the space of infinite sequence of symbols $A^\mathbb{N}$ from an countable alphabet $A$, then one can use the metric $\sum_{w \in A^*} 2^{-|w|} |\mu([w]) - \nu([w])|$, where $A^*$ is the set of finite words of $A$ and $|w|$ is the length of $w$, and $[w]$ is the cylinder set of all inifinite sequences starting with $w$. (The space $\{0,1\}^\mathbb{N}$ is often a useful approximation to $[0,1]$ where the words correspond to dyadic intervals.)
(You can ignore this...) Like I said, I work in computability theory and I find the weak convergence is the right topology to work with. Why? Well, separable topologies have a computability theoretic nature. In this case, given sufficiently good approximations to a measure in the weak topology, one can compute information about the measure. In particular, the weak topology gives exactly the information needed to compute $f \mapsto \int f\, d\mu$ for a bounded continuous function $f$. In the case of $A^\mathbb{N}$, it is exactly the information needed to compute $\mu{[w]}$ for each word $w \in A^*$. If I were to represent a distribution on a computer (with no worries about running times or memory usage) then I would represent it by approximating it by finite discrete measures which are close enough in some metric for weak convergence. (In the same way that reals are represented by rational approximations.)
