Question

What happens to the inequality sign of every component of a piecewise function f when we take its derivative?

Answer 1

what do you mean? how function can have inequality sign?

Answer 2

i mean expressions like this:

Answer 3

you would have to take the derivative of each of the possible fnctions, and they would apply to the same range of x, e.g. look at picture
simply, the inequalities stay the same