The definition of continuity in mathematics is the property that a function or curve is unbroken and connected without any abrupt changes or jumps. In other words, it means that a small change in the input of a function results in a small change in the output.
The three types of continuity are point continuity, interval continuity, and uniform continuity. Point continuity means that a function is continuous at a specific point, interval continuity means that a function is continuous over a specific interval, and uniform continuity means that a function is continuous over an entire domain.
Continuity and differentiability are closely related concepts in mathematics. A function is differentiable at a point if it is continuous at that point and has a well-defined derivative. In other words, differentiability is a stronger condition than continuity.
The inequality signs used to represent continuity are <, >, ≤, and ≥. These signs indicate the direction of the inequality and whether it includes or excludes the boundary points. For example, f(x) is continuous at x = a if and only if f(a+) = f(a) = f(a-).
A function is continuous if it satisfies the three criteria of continuity: the function is defined at the point, the limit of the function at the point exists, and the limit is equal to the value of the function at the point. In other words, a function is continuous if there are no breaks, holes, or jumps in its graph.