Types of Discontinuity

The continuity lesson gave a three-part test: defined, the limit exists, and the two are equal. When the test fails, the function has a discontinuity at , and the way it fails tells you which type. There are three main types of discontinuity, removable, jump and infinite, and one less common kind, oscillating. Telling them apart takes nothing more than the one-sided limits.

Removable Discontinuity

A removable discontinuity happens when exists, but is undefined or equal to something else. On a graph it’s a hole in an otherwise smooth curve, sometimes with a stray dot above or below it. It’s the only kind you can fix: redefine as the limit and the function becomes continuous.

Example. at . Factor: for . The limit is 5, but is undefined, so there’s a removable discontinuity at . Defining fills the hole.

In rational functions, removable discontinuities come from factors that cancel between the top and bottom. The rational functions lesson calls them holes.

Jump Discontinuity

A jump discontinuity happens when both one-sided limits are finite numbers but they’re different. The graph jumps from one height to another, and the size of the jump is the difference between the two limits.

Example. for and for . From the left, ; from the right, . The jump has size 3. Because the two sides disagree, no value of could make the function continuous, so a jump is non-removable.

Jumps are typical of piecewise functions and step functions such as the floor function, which jumps by 1 at every integer. They also model real prices: a parking fee that rises by a fixed amount at the start of each hour jumps at every hour mark.

Infinite Discontinuity

An infinite discontinuity happens when at least one one-sided limit is or . The graph has a vertical asymptote there, as covered in infinite limits.

Example. at : the left side falls to and the right side rises to , so there’s an infinite discontinuity. has one too, even though both sides go up.

Oscillating Discontinuity

Occasionally a function neither settles nor blows up. swings between and 1 faster and faster as , so neither one-sided limit exists. This is called an oscillating (or essential) discontinuity. It’s rare in practice, but it’s the reason the classification has more than three cases.

How to Find and Classify Points of Discontinuity

1. List the candidates. Zeros of every denominator, edges of the domain (from roots and logs), and the points where a piecewise function changes formula. Everywhere else, familiar functions are continuous.
2. Compute the one-sided limits at each candidate, and if it exists.
3. Classify with the flowchart: equal finite limits mean removable (unless they match , in which case it’s continuous after all), different finite limits mean a jump, and an infinite side means an infinite discontinuity.

Worked Example: Two Types in One Function

Find and classify the discontinuities of .

Factor: . The denominator is zero at and , so those are the candidates.

• At : the factor cancels, leaving . The limit exists but doesn’t, so this is a removable discontinuity, a hole at .
• At : the simplified form gives , so the one-sided limits are infinite. This is an infinite discontinuity, with the vertical asymptote .

This is the standard pattern for rational functions: factors that cancel leave holes, and factors that stay in the denominator leave asymptotes.

Removable vs Non-Removable

The practical difference is whether the limit exists. If it does, the discontinuity is removable and a single redefinition fixes it. If it doesn’t, because the sides disagree (a jump), blow up (infinite) or never settle (oscillating), the discontinuity is non-removable. When a question asks for “the value that makes continuous”, it’s always a removable discontinuity, and the answer is the limit.

Common Mistakes to Avoid

• Calling every zero of the denominator an asymptote. Cancel first; a cancelled factor gives a hole.
• Judging a jump from one side. You need both one-sided limits to see the jump.
• Thinking a dot fixes a jump. A filled dot at doesn’t help if the two sides approach different heights.
• Forgetting the switch points of a piecewise function. They’re the most common place for a jump.

Practice Problems

Work each one before opening the answer.

Problem 1. Classify the discontinuity of at .

Problem 2. Classify the discontinuity of at .

Problem 3. For when and when , classify the discontinuity at .

Problem 4. What value of removes the discontinuity of at ?

Problem 5. Find and classify every discontinuity of .

Next, the intermediate value theorem: why a continuous function can’t skip any values, and how to use that to prove an equation has a solution.