It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the function must be continuous at such a point p, in order to get the optimal linear approximation at that point.
I have two questions:
I am working with a nonlinear function in $R^3$ that is not continuous on the domain that I need the function approximated on; there is a "line" of discontinuity that splits the surface in two. Should I be using an 2-part piecewise function, where each equation is a linear approximation from the Jacobian?
If I'm not working with a dynamic system, should I not be constraining myself to using the Jacobian anyway?
Thank you in advance.
To have a reasonable linear approximation at $p$, the function must be differentiable at $p$; this is stronger than continuity. If your function is only piecewise differentiable, you can still use linear approximation away from the line where things break down. The approximation is local anyway. Yes, the derivative will be piecewise, like the function itself.
Question 2 is too vague to have an answer. Constrain yourself when it helps you focus; don't when it eliminates useful options...