What is define continuity?

In mathematics, continuity is a fundamental concept that describes the smoothness and unbroken nature of a function or a curve. The concept is closely related to limits and is used to describe whether a function has abrupt jumps or breaks or whether it can be drawn without lifting the pen.

A function is said to be continuous if its values change gradually from one point to another without any sudden changes or gaps. Mathematically, a function f(x) is said to be continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c).

Continuity is a crucial concept in calculus, where it is used to define derivatives and integrals. It is also an essential property in topology, where it is used to describe the shape and properties of geometric spaces. Many real-world problems can be modeled using functions, and continuity is an important property to ensure that these models are accurate and reliable.