I am working with some piecewise linear functions and its' intervals defined as $I_j$=($x_j$,$x_{j+1}$].
I want to use the derivatives of sub-functions as Lipschitz constants, but in order to do that, the sub-function must be continuous on [a,b] (according to mean value theorem).
So let's say a linear function is continuous and differentiable on (a,b], can I somehow represent its derivative as Lipschitz constant since it's not continuous on [a,b]?