A function is discontinuous at a point when you can’t draw through that point without lifting your pencil. More precisely, a function fails to be continuous when it breaks one of three requirements: the function must have a value at the point, the limit must exist at the point, and the limit must equal the function’s value. If any one of these conditions fails, you have a discontinuity.
The Three Conditions for Continuity
To understand what makes a function discontinuous, start with what makes it continuous. A function f(x) is continuous at a point x = a when all three of these hold:
- f(a) is defined. The function actually produces a value at that point. There’s no hole or gap in the domain.
- The limit of f(x) as x approaches a exists. As you get closer and closer to a from both sides, the function values settle toward a single number.
- The limit equals f(a). The value the function is heading toward actually matches the value it takes when it gets there.
A discontinuity happens whenever at least one of these conditions breaks down. Each type of failure produces a different kind of discontinuity, and recognizing which condition failed tells you exactly what type you’re dealing with.
Removable Discontinuities (Holes)
A removable discontinuity is the mildest type. The limit exists, but either the function isn’t defined at that point or it’s defined as the “wrong” value. On a graph, this looks like a hole: a tiny open circle where a point should be.
The most common way this happens is with rational functions that have a common factor in the numerator and denominator. Take f(x) = (x − 1)(x − 2) / (x − 1)(x + 4). The factor (x − 1) appears in both the top and bottom, which means the function is undefined at x = 1 because you’d be dividing by zero. But if you cancel that common factor, the rest of the function behaves perfectly fine near x = 1. The limit exists; the function just doesn’t have a value there. You could “remove” the discontinuity by filling in that hole with the right value.
Piecewise functions can also create removable discontinuities. If one piece of the function assigns an unusual value at a single point, the limit from both sides might agree on one number while the function itself takes a different value. The third condition (limit equals function value) fails, but you could fix it by redefining that one point.
To find removable discontinuities in a rational function, factor the numerator and denominator completely, then look for factors that appear in both. Set each common factor equal to zero and solve. Those x-values are your holes.
Jump Discontinuities
A jump discontinuity happens when the function suddenly leaps from one height to another, like a step in a staircase. The left-hand limit (approaching from smaller x-values) and the right-hand limit (approaching from larger x-values) both exist and are finite, but they aren’t equal. Because the two sides don’t agree, the overall limit doesn’t exist, and the second condition for continuity fails.
Piecewise functions are the classic source of jump discontinuities. Imagine a function defined as one formula for x < 2 and a different formula for x ≥ 2. If the first formula gives a value of 5 as x approaches 2 from the left, but the second formula starts at 8 when x = 2, the graph has a visible break. No amount of redefining a single point can fix this, because the gap between the two sides is the problem.
On a graph, you’ll typically see two curve segments that end at different heights, often marked with a closed dot on one end and an open dot on the other to show which value the function actually takes.
Infinite Discontinuities (Vertical Asymptotes)
An infinite discontinuity occurs when the function’s values blow up toward positive or negative infinity as x approaches a particular point. This is the behavior you see at a vertical asymptote. The function f(x) = 1/x, for instance, shoots toward positive infinity as x approaches 0 from the right and toward negative infinity from the left.
Technically, if either the left-hand or right-hand limit heads toward infinity, the limit doesn’t exist in the usual sense, so condition two fails. Unlike a removable discontinuity, there’s no finite value you could assign to fix the problem. The function is genuinely undefined there because division by zero (or a similar operation) makes it impossible to produce a real number.
Rational functions produce vertical asymptotes at x-values that make the denominator zero but don’t cancel with the numerator. This is the key distinction from removable discontinuities: if the zero in the denominator has no matching factor in the numerator, you get an asymptote instead of a hole.
Oscillating Discontinuities
The most exotic type is the oscillating discontinuity, where the function bounces back and forth so wildly near a point that it never settles on a limit. The standard example is f(x) = sin(1/x) at x = 0. As x gets closer to zero, 1/x grows without bound, and the sine function cycles faster and faster between −1 and 1. The oscillations become infinitely rapid, so the limit simply doesn’t exist.
This is different from an infinite discontinuity because the function values stay bounded (they never go to infinity). It’s also different from a jump, because there aren’t just two competing values. The function takes on every value between −1 and 1 infinitely many times in any interval around x = 0. There’s no way to redefine f(0) to make the function continuous, which is why this falls under the broader category sometimes called “essential” discontinuities.
How to Check a Piecewise Function for Discontinuities
When you’re given a piecewise function, the potential trouble spots are the boundary points where the formula changes. At each boundary x = c, work through three steps:
First, plug c into whichever piece of the function covers that point and confirm you get a real number. This checks that f(c) is defined. Second, compute the left-hand limit by plugging c into the formula that applies just to the left of c, and compute the right-hand limit using the formula just to the right. If these two limits aren’t equal, you have a jump discontinuity and can stop. Third, if the left and right limits do match, compare that shared limit value to f(c). If they’re equal, the function is continuous at that point. If not, you have a removable discontinuity.
For example, suppose a function uses 2x + 1 for x < 3 and x² − 2 for x ≥ 3. The left-hand limit as x approaches 3 is 2(3) + 1 = 7. The right-hand limit is 3² − 2 = 7. Both sides agree, and f(3) = 7 from the second piece, so this function is actually continuous at x = 3. If the second piece had been x² instead, f(3) would equal 9, the limits would disagree, and you’d have a jump.
Spotting Discontinuities on a Graph
Each type of discontinuity has a distinct visual signature. A removable discontinuity appears as a small open circle (hole) on an otherwise smooth curve. Sometimes there’s a separate filled dot at a different height, indicating the function was assigned a mismatched value at that point.
Jump discontinuities show two curve segments ending at different heights, with the gap between them representing the “jump.” You’ll see one closed dot and one open dot at the boundary, or two separate branches that simply don’t connect.
Infinite discontinuities are the most dramatic visually. The curve swoops upward or downward toward a vertical asymptote, with the function values growing without bound. The graph never touches or crosses the vertical line at the discontinuity.
Oscillating discontinuities are hard to graph accurately because the function vibrates infinitely fast. Near the trouble point, any plot will look like a dense, tangled mess of oscillations that get increasingly compressed.
Discontinuity vs. Not in the Domain
One subtle point worth clarifying: a discontinuity is specifically a problem at a point that’s relevant to the function’s domain. If a point was never in the function’s domain to begin with, some textbooks don’t consider it a true discontinuity. For instance, the square root function isn’t defined for negative numbers, but we don’t typically call every negative number a discontinuity. A discontinuity implies the function is defined at or very near the point, and something goes wrong with the limit or the value there. The distinction matters more in formal proofs than in a calculus class, but it helps to know that “undefined” and “discontinuous” aren’t always the same thing.