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In simple terms, a continuous function is one whose graph can be drawn without lifting your pen from the paper. There are no abrupt jumps, breaks, or holes.
More rigorously, it means that small changes in the input of a function result in small changes in the output.
Formal Definition:
In calculus and mathematical analysis, the concept is defined precisely using limits. A function f(x) is continuous at a point 'a' if:
f(a) is defined.
The limit of f(x) as x approaches 'a' exists.
The limit of f(x) as x approaches 'a' is equal to f(a).
Importance in University Math:
Continuity is a fundamental concept in calculus, real analysis, and related fields. It's crucial for understanding:
Limits and derivatives: The definition of a derivative relies on the concept of limits, which is closely tied to continuity.
Integrals: The definite integral of a continuous function is well-defined.
Theorems: Many important theorems in calculus, such as the Intermediate Value Theorem and the Extreme Value Theorem, rely on the continuity of functions.
Real analysis: a large part of real analysis is based on rigorous definitions of continuity, and theorems relating to continuous functions.
How Continuity Fits into University-Level Math:

Calculus:
Students encounter continuity early in calculus courses. They learn to identify continuous functions, find points of discontinuity, and apply theorems related to continuity.
Real Analysis:
In real analysis, the concept of continuity is explored in greater depth, with rigorous definitions and proofs. Students learn about different types of continuity and explore the properties of continuous functions on various domains.