Limits and Continuity

Limits

The concept of a limit is fundamental in calculus and analysis. Informally, a limit is the value that a function or sequence "approaches" as the input or index approaches some value.

For a function f(x) and a point c in its domain, the limit of f(x) as x approaches c is represented as lim_{x→c} f(x). If f(x) approaches some real number L as x gets closer and closer to c, then we write lim_{x→c} f(x) = L.

Limits can also be defined for points not in the domain of f(x), at infinity, or for sequences of real numbers.

Continuity

A function f(x) is said to be continuous at a point c in its domain if lim_{x→c} f(x) = f(c). In other words, a function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point.

A function f(x) is said to be continuous on an interval if it is continuous at every point in that interval.

Importance in Analysis

The concepts of limits and continuity are used throughout mathematical analysis. They form the foundation for the definition of the derivative and the integral in calculus. They also play a key role in the study of sequences and series, and in the definition and properties of various types of functions.

Understanding limits and continuity is crucial for understanding more advanced concepts in mathematical analysis, such as differentiability, integrability, and convergence.