Continuity- Definition, Key Points, Notes 

Continuity is a fundamental concept in calculus that describes the smoothness and connectedness of a function. A function is said to be continuous if, intuitively, it can be drawn without lifting the pen from the paper.

Formally, a function f(x) is continuous at a point c if three conditions are satisfied:

1. f(c) is defined, meaning that the function has a value at c.
2. The limit of f(x) as x approaches c exists.
3. The limit of f(x) as x approaches c is equal to f(c).

In other words, a function is continuous if there are no sudden jumps, holes, or vertical asymptotes in its graph. It implies that the function can be evaluated and graphed without any abrupt changes or discontinuities.

Continuity is an essential property for various mathematical applications, including differentiation and integration. It allows for the use of powerful techniques and theorems in calculus, such as the Intermediate Value Theorem and the Mean Value Theorem.