I am currently trying to understand the proof for the boundedness of the Delta of an Option that was done in the paper "General Properties of Option Prices" (1996, Yaacov Z. Bergman, Bruce D. Grundy, Zvi Wiener). While I understand the proof in the case where the payoff function is differentiable, I am struggling to understand the generalization of it. There, the authors prove the boundedness of the Delta when the payoff function only has left and right derivatives everywhere ("where the two need not be equal, and where one of them may be plus infinity... or minus infinity", p. 1600).
This is what was written for the case where the payoff function is differentiable: Original Theorem + Proof
While this is what the Authors write for the generalization: Generalized Theorem + Proof
In the proof of the generalized Theorem they used the "generalization of the Intermediate Value Theorem of differential calculus to the case where only the left and the right derivatives are guaranteed to exist". I have not found such a generalization of the IVT so far and am wondering where this is stated.