The Fermat, Roll and Lagrange theorems are generalized into the class of nondifferentiable functions, the necessary and sufficient conditions for monotonicity and convexity of one variable functions are given.
Alferov, G., Ivanov, G., Sharlay, A., & Fedorov, V. (2019). Application of derived numbers theory in convex analysis. In AIP Conference Proceedings (Vol. 2116). American Institute of Physics Inc. https://doi.org/10.1063/1.5114063