Continuity is a fundamental concept in calculus that describes the smoothness of a function. A function is considered continuous at a point if its limit exists at that point, and the limit is equal to the function’s value at that point. In other words, a continuous function has no breaks, gaps, or jumps in its graph.
Mathematically, a function f is continuous at a point a if the following three conditions are satisfied:
- The function f is defined at a.
- The limit of f as x approaches a exists.
- The limit of f as x approaches a is equal to f(a).
Continuity plays a crucial role in calculus, as many theorems and techniques rely on the continuous nature of functions. For example, the Intermediate Value Theorem states that if a function is continuous on a closed interval, then it takes on every value between its endpoints. Similarly, the Mean Value Theorem and the Fundamental Theorem of Calculus require the continuity of functions for their validity.