To find a jump discontinuity, you evaluate the left-hand limit and the right-hand limit of a function at a given point. If both limits exist as finite numbers but are not equal to each other, that point is a jump discontinuity. This is the single test that defines it, and once you understand the mechanics, spotting jumps becomes straightforward.
What Makes a Discontinuity a “Jump”
A jump discontinuity occurs at a point where the function abruptly shifts from one value to another with no smooth transition in between. Mathematically, three conditions must all be true at a point x = c:
- The left-hand limit exists and is finite. As x approaches c from smaller values, the function settles toward a specific number.
- The right-hand limit exists and is finite. As x approaches c from larger values, the function settles toward a different specific number.
- Those two limits are not equal. The mismatch is what creates the “jump.”
It doesn’t matter what the function’s actual value is at x = c, or even whether the function is defined there at all. The jump is entirely about the disagreement between the two one-sided limits.
Step-by-Step Process
Start by identifying where the function changes its rule. Most jump discontinuities occur in piecewise functions, where a formula switches at a boundary point. That boundary is your candidate.
Next, compute the left-hand limit. Plug values approaching the candidate point from the left side into the formula that governs that region, and evaluate the limit. Then compute the right-hand limit using the formula that applies on the other side. Compare the two results. If both are finite real numbers and they differ, you’ve confirmed a jump discontinuity.
Here’s a concrete example. Suppose a function is defined as f(x) = -x² + 16 when x < 0, and f(x) = x + 8 when x ≥ 0. To check x = 0:
- Left-hand limit: As x approaches 0 from the negative side, use -x² + 16. That gives -(0)² + 16 = 16.
- Right-hand limit: As x approaches 0 from the positive side, use x + 8. That gives 0 + 8 = 8.
Since 16 ≠ 8, there is a jump discontinuity at x = 0. The size of the jump is the absolute difference between the two one-sided limits: |16 – 8| = 8.
How It Differs From Other Discontinuities
Not every break in a function is a jump. There are three main types of discontinuity, and they each fail the continuity test in a different way.
A removable discontinuity is a “hole” in the graph. The left-hand and right-hand limits agree with each other, but the function either isn’t defined at that point or is defined as the wrong value. You could fix it by redefining a single point. A jump can’t be fixed this way because the two sides of the function genuinely disagree.
An infinite discontinuity is what you see at a vertical asymptote. At least one of the one-sided limits shoots off toward positive or negative infinity rather than settling on a finite number. The key distinction is that jump discontinuities require both one-sided limits to be finite. If either limit is infinite, it’s not a jump.
Where Jump Discontinuities Show Up
Piecewise-defined functions are the most common source in a calculus course, but they also appear naturally in several standard functions.
The floor function (written ⌊x⌋), which rounds any number down to the nearest integer, has a jump discontinuity at every integer. At x = 3, for instance, the left-hand limit is 2 (the function outputs 2 for all values just below 3), while the right-hand limit is 3. Each jump has a size of exactly 1. The ceiling function (⌈x⌉), which rounds up, behaves the same way, with unit jumps at every integer.
The Heaviside step function is another classic example. It equals 0 for all negative inputs and 1 for all positive inputs, creating a single jump of size 1 at x = 0. This function is widely used in physics and engineering to model anything that switches on or off instantaneously, like a circuit closing or a signal activating.
Reading a Jump on a Graph
On a graph, a jump discontinuity looks like two separate curve segments that end at different heights above the same x-value. The standard convention uses circles to mark these endpoints. A solid (filled) circle means the function is actually defined at that point and takes that value. An open (hollow) circle means the curve approaches that point but the function does not equal that value there.
At a jump, you’ll typically see one solid circle and one open circle stacked vertically, or two open circles if the function is defined as yet a third value at that point (or not defined at all). The vertical gap between the two circles represents the size of the jump. If you’re graphing a piecewise function yourself, always check which piece includes the boundary point (look for ≤ vs. < in the domain conditions) and place your solid circle on that piece.
Tips for Tricky Cases
When a piecewise function has more than two pieces, check every boundary where the formula changes. A function could be continuous at some boundaries and have jumps at others. Evaluate each one independently.
Absolute value functions and functions involving square roots can sometimes be rewritten as piecewise functions. If you suspect a discontinuity but the function isn’t written in piecewise form, try splitting it at the point where the expression inside changes behavior (where the argument of the absolute value equals zero, for example) and then check the one-sided limits.
Functions with jump discontinuities are not differentiable at the jump. You cannot take a standard derivative there. In advanced mathematics, the derivative of the Heaviside step function is treated as the Dirac delta function, a theoretical object rather than a regular function. For a standard calculus course, it’s enough to know that a jump kills differentiability at that point, even though the function remains differentiable everywhere else.
Finally, remember that the size of a jump is always measured as the absolute difference between the right-hand limit and the left-hand limit. This quantity is sometimes called the “jump size” or “saltus.” In the floor function, every jump has size 1. In a custom piecewise function, the size depends on the specific formulas involved, and computing it is as simple as subtracting the two one-sided limits once you’ve found them.