  The paper deals with variational approaches to the segmentation of time
series into smooth pieces, but allowing for sharp breaks. In discrete time, the
corresponding functionals are of Blake-Zisserman type. Their natural
counterpart in continuous time are the Mumford-Shah functionals. Time series
which minimise these functionals are proper estimates or representations of the
signals behind recorded data. We focus on consistent behaviour of the
functionals and the estimates, as parameters vary or as the sampling rate
increases. For each time continuous time series $f\in L^2 (\lbrack 0,1\rbrack)$
we take conditional expectations w.r.t. to $\sigma$-algebras generated by finer
and finer partitions of the time domain into intervals, and thereby construct a
sequence $(f_n)_{n\in\N}$ of discrete time series. As $n$ increases this
amounts to sampling the continuous time series with more and more accuracy. Our
main result is consistent behaviour of segmentations w.r.t. to variation of
parameters and increasing sampling rate.
