Continuity

Continuity

Definition:

  • Continuity of a Function: A function f(x) is continuous at a point c in its domain if the following conditions hold:
    1. f(c) is defined.
    2. The limit of f(x) as x approaches c exists.
    3. The limit of f(x) as x approaches c equals f(c).

Types of Continuity:

  • Point Continuity: A function is continuous at a specific point c if limxcf(x)=f(c).

  • Interval Continuity: A function is continuous on an interval if it's continuous at every point within that interval.

Types of Discontinuities:

  • Removable Discontinuity: A discontinuity that can be "filled in" or removed by redefining the function at that point.

  • Jump Discontinuity: When the function changes abruptly from one value to another at a specific point.

  • Infinite Discontinuity: Occurs when the function's value becomes unbounded (tends to infinity) at a certain point.

  • Essential Discontinuity: A discontinuity that can't be classified as removable, jump, or infinite; often involves oscillatory behavior.

Properties of Continuous Functions:

  • Algebra of Continuous Functions: The sum, difference, product, and quotient of continuous functions are continuous.

  • Intermediate Value Theorem: If f(x) is continuous on an interval [a,b] and k lies between f(a) and f(b), then there exists c in (a,b) such that f(c)=k.

Testing Continuity:

  • Direct Substitution: Checking if the function is defined at a given point.

  • Using Limits: Evaluating limits as x approaches a value to verify if the function's values approach a specific value.

  • Graphical Analysis: Observing the graph for any breaks, jumps, or asymptotic behavior.