if you thought it likely that the underlying phenomenon has discontinuities, and you didn't want your model to smooth them out

This is a change point problem. See the example in section 3.1 of the PyMC manual:

Consider the following dataset, which is a time series of recorded coal mining disasters in the UK from 1851 to 1962 [. . . .] Occurrences of disasters in the time series is thought to be derived from a Poisson process with a large rate parameter in the early part of the time series, and from one with a smaller rate in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations.

Taking averages is sure to smooth out discontinuities in the true values, isn't it?

Yes. If the true change points are unknown, then even if every possible underlying phenomenon has discontinuities, the average of credible underlying phenomena (the posterior mean) can still be continuous. See this plot in the Introduction to Bayesian Thinking blog post "A Poisson Change-Point Model" (discontinuous mining safety regulations) and Figure 5 on page 8 of "Bayesian change-point analyses in ecology" by Brian Beckage et al. (discontinuous border between canopy and gap conditions).