####################################
##┌─────────────────┬────┐#####
##│######################│#####└──┐##
##│######################│#########│##
##│######################│#####───┤##
##│######################│####### ┌┘##
##│###########┌─────┐ ##│###┌──┘###
##│##########┌┘#### # └──┴──┘######
##└─┐######┌─┘######################
####└──────┘########################

⣰⣾⣿⣶⣄⠄⠄⠄⠄⢀⣠⣤⡀⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄ ⠻⣿⣿⣿⠟⠄⠄⠄⢰⣿⣿⣿⣿⡆⠄⠄⠄⠄⠄⠄⠄⣀⠄⠄⠄⠄⠄⠄ ⠄⢈⣭⣤⡀⠄⢀⣤⣤⣛⠿⠿⠟⠄⠄⠄⠄⠄⠄⣰⣿⣿⣿⣄⠄⠄⠄⠄ ⠄⣿⢷⣾⢻⡀⣿⣱⣮⢿⣷⣤⡀⠄⠄⢀⣀⣀⣀⠹⣿⣿⣿⠟⠄⠄⠄⠄ ⠄⣿⣧⢿⣷⣳⣿⣧⢿⣷⡘⠻⣿⡿⠿⢿⣿⢿⣿⢿⣤⡀⠄⠄⠄⠄⣀⡀ ⠄⢻⣿⣯⣿⢿⣾⣯⣏⣛⡿⣷⣶⣤⣔⢿⣿⣯⢿⣯⡙⢿⣷⣾⠿⠿⠟⠋ ⠄⠈⠿⣿⣿⣷⠄⠿⣝⣛⣦⢰⣾⣯⣯⣾⡿⠏⠄⢻⣷⣦⣄⣀⣀⠄⠄⠄ ⠄⠄⠄⠘⣿⣿⡆⠄⠸⣿⣿⣄⠙⢿⣿⣷⣀⠄⠄⠄⠉⠉⠙⠛⠛⠛⠄⠄ ⠄⠄⠄⠄⠹⣿⣿⡄⠄⢻⣿⣿⡄⠄⠙⣿⣿⣷⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄ ⠄⠄⠄⠄⣼⣿⣿⠁⠄⣼⣿⣿⠄⠄⢠⣿⣿⠟⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄ ⠄⠄⠄⠄⣿⣿⡇⠄⢠⣿⣿⡏⠄⣴⣿⣿⠋⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄

⠀⠀⠀⠀⠀⣿⣧⣄⠀⣤⡤⠀⠀⠀⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣯⡜⠀⠈⠓⠂⠀⢰⣿⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣧⣤⡄⠀⠠⣠⣾⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡤⢤⢶⣿⣿⡞⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠻⢻⠟⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⡿⠋⠁⣠⣾⣿⣦⣄⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀
⠀⠤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⠀⢰⡖⠂⡆⠀⡆⢠⣒⣲⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⠀⠸⠇⠀⠳⠤⠃⠘⠦⠤⠀⠶⠀⠆⠀⠆⠀

⢸
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣄ <- Matute
⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠹⣿⣧⣀⣠⣴⣾⣷⣿⣷⠾⢷⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡷
⠀⠀⠈⢿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⢿⣿⣄⠀⠀⠀⠀⢠⣶⣾⣿⡇ <- Shackmerferu
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠙⠿⡿⢆⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣤⣶⣾⣿⣿⣿⣿⣷⠹⣷⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢀⣾⣿⣿⣿⣿⣏⠀⠀⢀⣀⣈⣉⣉⣉⣙⣁⣀
⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣷⠈⠉⠙⠛⢻⣭⣷
⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠘⢿⡿⠇⠻⣿⠟⠀⠀⠀⠀⢿⣿⠇

The fundamental theorem of calculus provides a generic formulation of the inverse relationship between the theories of differentiation and integration; meaning,which is satisfied in Cauchy integration theory. However, the fundamental theorem of calculus has no meaning if the derivative 𝑓´(𝑥) of a differentiable function is not Riemann integrable. The purpose of the research is to preset the proprieties which should fulfill the derivative 𝑓´(𝑥) of a differentiable function in order to satisfy the Fundamental Theorem of Calculus in the Riemann integration theory.

⣿⡇⣿⣿⣿⠛⠁⣴⣿⡿⠿⠧⠹⠿⠘⣿⣿⣿⡇⢸⡻⣿⣿⣿⣿⣿⣿⣿
⢹⡇⣿⣿⣿⠄⣞⣯⣷⣾⣿⣿⣧⡹⡆⡀⠉⢹⡌⠐⢿⣿⣿⣿⡞⣿⣿⣿
⣾⡇⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣦⡀⠁⢸⡌⠻⣿⣿⣿⡽⣿⣿
⡇⣿⠹⣿⡇⡟⠛⣉⠁⠉⠉⠻⡿⣿⣿⣿⣿⣿⣦⣄⡉⠂⠈⠙⢿⣿⣝⣿
⠤⢿⡄⠹⣧⣷⣸⡇⠄⠄⠲⢰⣌⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠄⠈⠻⢮
⠄⢸⣧⠄⢘⢻⣿⡇⢀⣀⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠄⢀
⠄⠈⣿⡆⢸⣿⣿⣿⣬⣭⣴⣿⣿⣿⣿⣿⣿⣿⣯⠝⠛⠛⠙⢿⡿⠃⠄⢸
⠄⠄⢿⣿⡀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡾⠁⢠⡇⢀
⠄⠄⢸⣿⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣫⣻⡟⢀⠄⣿⣷⣾
⠄⠄⢸⣿⡇⠄⠈⠙⠿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⠊⢀⡇⣿⣿
⠒⠤⠄⣿⡇⢀⡲⠄⠄⠈⠙⠻⢿⣿⣿⠿⠿⠟⠛⠋⠁⣰⠇⠄⢸⣿⣿⣿