This thesis explores the intricate relationship between differentiability and monotonicity in real functions, two fundamental concepts in mathematical analysis. Differentiability, indicating the presence of a derivative at every point, and monotonicity, describing a function's consistent increase or decrease within an interval, are pivotal in understanding the behavior of functions across various domains of mathematics. This research seeks to deepen the theoretical understanding of these unique mathematical behaviors by investigating the conditions under which such functions exist and characterizing their properties through analytical methods.