Fitting data with piecewise models given fixed points of fusage, or intervals of validity with given number of continous moments across the models is a well explored problem with hundreds if not thousand of methods. But what if we don't know the points of fusage of the models? Can we find a method to adaptively find the points of fusage or intervals of validity for each model?
Here is example of a noisy function which we want to fit say two polynomials $p_1(x),p_2(x)$ to. How can we choose the fuseagepoint $x_0$ to get as good a fit as possible given we enforce $$p_1^{(k)}(x_0) = p_2^{(k)}(x_0), \forall k \in [0,n]$$
or a softer variant, penalty terms added to the piecewise fitter (assuming an optimization):
$$\sum_k\epsilon_k\|p_1^{(k)}(x_0) - p_2^{(k)}(x_0)\|_2^2$$
