Discontinuity

Definition

Discontinuity refers to a break or interruption in the continuity of a function. It occurs when a function is not defined at a particular point or when the function exhibits a sudden jump or change in its value at a specific point within its domain.

5 Must Know Facts For Your Next Test

1. Discontinuities can occur in both rational functions and trigonometric functions, affecting the behavior and graphical representation of these functions.
2. Identifying and classifying the type of discontinuity is crucial for understanding the properties and behavior of a function.
3. Rational functions, which are defined as the quotient of two polynomial functions, can exhibit removable, jump, or infinite discontinuities.
4. Trigonometric functions, such as the secant, cosecant, and cotangent functions, can also display discontinuities at specific points within their domains.
5. The presence of discontinuities in a function can significantly impact its graphical representation, limiting the function's domain and affecting its overall behavior.

Review Questions

• Explain how discontinuities can affect the behavior of rational functions.
  • Discontinuities in rational functions can have a significant impact on the function's behavior. Removable discontinuities can be eliminated by redefining the function at the point of discontinuity, while jump discontinuities result in sudden changes or jumps in the function's value. Infinite discontinuities, where the function approaches positive or negative infinity, can create vertical asymptotes in the function's graph, which are important features to understand when analyzing the properties and behavior of rational functions.
• Describe the role of discontinuities in the graphs of the other trigonometric functions, such as secant, cosecant, and cotangent.
  • The graphs of the secant, cosecant, and cotangent functions can exhibit discontinuities at specific points within their domains. These discontinuities occur at the values where the denominator of the function becomes zero, resulting in vertical asymptotes in the function's graph. Understanding the locations and behavior of these discontinuities is crucial for accurately sketching and interpreting the graphs of the other trigonometric functions, as they can significantly impact the function's overall shape and properties.
• Analyze how the presence of discontinuities in a function can influence its domain and the overall behavior of the function.
  • Discontinuities in a function can have a profound impact on its domain and overall behavior. The presence of a discontinuity may limit the function's domain by excluding the point(s) of discontinuity. Additionally, the type of discontinuity (removable, jump, or infinite) can affect the function's behavior, such as causing sudden changes, vertical asymptotes, or limiting the function's range. Carefully identifying and classifying the discontinuities in a function is essential for understanding its properties, sketching its graph accurately, and analyzing its behavior within the given domain.

Related terms

Removable Discontinuity: A discontinuity that can be eliminated by redefining the function at the point of discontinuity, without changing the function's behavior elsewhere.

Jump Discontinuity: A discontinuity where the function exhibits a sudden jump or change in its value at a specific point within its domain.

Infinite Discontinuity: A discontinuity where the function approaches positive or negative infinity at a specific point within its domain.