Abstract: The function is a basic concept in mathematics, and continuity is one of its important properties. The concept of a function’s continuity stems from geometric intuition. Because a function was defined as analytical expressions, a continuous function was initially seen as one determined by a single analytical law. Following the trend towards rigor in analysis, Cauchy gave the concept of a function’s continuity in the modern sense by formulating the geometric notion of continuity. Based on Cauchy’s work, Weierstrass gave the current definition of a function’s continuity. The historical exploration on the evolution of the concept of a function’s continuity can provide a window into the rigorous process of analysis.

Key Words: Rigor in analysis; Function; Continuity; Differentiability; Integrability