Antiderivative of a branch function
Let , for and for . I want to determine the real numbers a and b for which f admits antiderivatives.
I took a plausible antiderivative for , and for and using the fact that F must be first continuous and then differentiable, I showed that .
This is easy to see as from continuity we must have that .
Then, since F is clearly differentiable on if we impose the condition that the left and right derivatives at 0 of F to be equal so we must have:
Which gives us .
My question is then, does this suffice? Can a be chosen arbitrarily such that f has antiderivatives as long as . It would seem that the function for and for for does indeed satisfy the necessary conditions.