Types of discontinuity
A removable discontinuity is a single hole. A jump has different left and right limits. An infinite discontinuity has the function growing without bound.
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Analogy
Three kinds of road damage: a small pothole, a step up, and a cliff edge.
Example
(x^2-1)/(x-1) at 1: removable hole.|x|/x at 0: jump, from -1 to 1.1/x^2 at 0: infinite, grows without bound.
In practice
Naming the type tells you whether the function can be repaired by redefining one value.
Practice
Flashcards
- Which discontinuity has different left and right limits?A jump discontinuity.
Interview questions
- Describe the three types of discontinuity and give an example of each.
- Removable: hole, limit exists, value missing
- Jump: left and right limits differ
- Infinite: function grows without bound
- Examples such as (x^2-1)/(x-1), |x|/x, 1/x^2
Questions people ask
Can a removable discontinuity be fixed?
Yes. If the limit exists but f(a) is missing or wrong, you define a new function with f(a) set equal to the limit. For (x^2 - 1)/(x - 1), setting the value at 1 to 2 fills the hole and makes the function continuous.
What is a jump discontinuity?
A jump happens when the left-hand and right-hand limits both exist but are different numbers. The graph suddenly steps from one height to another. The sign function, which gives -1 for negative inputs and 1 for positive inputs, jumps by 2 at zero.