Continuity

In mathematics, continuity refers to the smoothness or uninterrupted nature of a function. A function is said to be continuous if it can be drawn without lifting the pen from the paper. This means that there are no breaks, jumps, or holes in the graph of the function.

Key Concepts:

1. Continuity of a Function: A function f(x) is continuous at a specific point c if the following conditions are met:
   • f(c) is defined (i.e., the function exists at c)
   • The limit of f(x) as x approaches c exists
   • The limit of f(x) as x approaches c is equal to f(c)
2. Types of Discontinuities: There are several types of discontinuities, including:
   • Removable Discontinuity
   • Jump Discontinuity
   • Infinite Discontinuity
3. Continuity on an Interval: A function is said to be continuous on a closed interval [a, b] if it is continuous at every point in the interval and at the endpoints a and b.

Study Guide:

Here are some key points to remember when studying continuity:

1. Understand the concept of continuity and what it means for a function to be continuous at a point.
2. Learn how to identify different types of discontinuities and understand their graphical representations.
3. Practice determining the continuity of a function on a given interval by checking the conditions for continuity at each point in the interval.
4. Work on problems involving limits to strengthen your understanding of continuity and its relationship to the behavior of functions.
5. Use graphing tools to visualize the continuity of functions and identify points of discontinuity.

Remember to practice solving problems related to continuity to reinforce your understanding of the concept.