Exploring Types of Discontinuities

students, imagine a road on a GPS map 🚗. Most of the time the road is smooth, but sometimes there is a gap, a detour, or a sudden cliff. In calculus, those “breaks” are called discontinuities. Understanding them is essential because limits and continuity tell us how a function behaves near a point, even when the function itself may not be perfectly defined there.

In this lesson, you will learn how to identify the main types of discontinuities, describe what each one means, and connect these ideas to AP Calculus AB skills. By the end, you should be able to explain why a function fails to be continuous, use limits to classify that failure, and recognize how these ideas fit into the bigger picture of limits and continuity.

What continuity means before we look at breaks

A function is continuous at a point $x=c$ when three things happen together:

1. $f(c)$ is defined.
2. $\lim_{x\to c} f(x)$ exists.
3. $\lim_{x\to c} f(x)=f(c)$.

If any one of these fails, the function is discontinuous at $x=c$.

That simple idea is powerful. It means continuity is not just about whether a graph can be drawn without lifting your pencil ✏️. It is also about whether the function has a value at the point and whether the nearby behavior matches that value. A graph can look almost smooth but still have a break in one of these conditions.

For AP Calculus AB, this matters because limits often describe what happens “near” a point, while continuity describes what happens “at” the point. A discontinuity is where those ideas do not line up.

Removable discontinuities: a hole in the graph

A removable discontinuity happens when a function is undefined at a point, or defined in a way that does not match the limit, but the limit still exists. The name “removable” comes from the fact that the function can often be fixed by redefining the value at that point.

A common example is

$$f(x)=\frac{x^2-1}{x-1}$$

for $x\neq 1$.

This expression simplifies to

$$f(x)=x+1$$

for $x\neq 1$.

The limit as $x\to 1$ is

$$\lim_{x\to 1} \frac{x^2-1}{x-1}=2.$$

But $f(1)$ is not defined in the original formula. So the graph has a hole at $x=1$. If we define $f(1)=2$, the function becomes continuous there.

This type of discontinuity often appears after simplifying algebraic expressions. In many AP questions, factoring and canceling a common factor reveals the hidden hole. The key idea is that the limit exists, but the point itself is missing or mismatched.

Real-world connection 🌱: Suppose a website calculator gives a strange blank result at one input even though nearby inputs work fine. If the system can be corrected by assigning the missing value, that is like a removable discontinuity.

Jump discontinuities: when left and right do not meet

A jump discontinuity occurs when the left-hand limit and right-hand limit both exist, but they are not equal.

That means

$$\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x).$$

Since the two one-sided limits are different, the two-sided limit does not exist. The graph “jumps” from one height to another.

For example, consider a piecewise function such as

$$f(x)=\begin{cases}

1, & x<0 \\

$3, & x\ge 0$

$\end{cases}$$$

At $x=0$,

$$\lim_{x\to 0^-} f(x)=1$$

and

$$\lim_{x\to 0^+} f(x)=3.$$

Because these are not equal, $\lim_{x\to 0} f(x)$ does not exist.

Jump discontinuities are common in situations where something changes suddenly, like tax brackets, shipping fees, or a plan that costs one amount up to a limit and another amount after that. The graph may be perfectly defined at the point, but the sudden change means it is not continuous there.

On AP Calculus AB, you should be able to identify jump discontinuities from graphs, tables, and formulas. One useful strategy is to compare the values approached from the left and right. If they disagree, you have a jump.

Infinite discontinuities: the function grows without bound

An infinite discontinuity occurs when the values of the function increase or decrease without bound near a point. In symbol form, this can look like

$$\lim_{x\to c} f(x)=\infty$$

or

$$\lim_{x\to c} f(x)=-\infty.$$

A classic example is

$$f(x)=\frac{1}{x-2}.$$

As $x\to 2^+$, the function increases without bound:

$$\lim_{x\to 2^+} \frac{1}{x-2}=\infty.$$

As $x\to 2^-$, it decreases without bound:

$$\lim_{x\to 2^-} \frac{1}{x-2}=-\infty.$$

This creates a vertical asymptote at $x=2$.

An important AP idea is that infinite discontinuities are connected to vertical asymptotes, but not every discontinuity is infinite. Some are holes, and some are jumps. The graph of an infinite discontinuity gets closer and closer to a vertical line while never crossing in a finite way near the asymptote.

Real-world connection ⚠️: Imagine a formula that models cost per item when dividing by the number of items ordered. If the denominator gets very close to $0$, the cost may become extremely large. That kind of blow-up behavior is similar to an infinite discontinuity.

Discontinuities at endpoints and on intervals

Continuity can also be discussed on intervals, not just at single points. A function is continuous on an interval if it is continuous at every point in that interval.

For a closed interval $[a,b]$, continuity at the endpoints is checked using one-sided limits:

• At $x=a$, use $\lim_{x\to a^+} f(x)$.
• At $x=b$, use $\lim_{x\to b^-} f(x)$.

This matters because there is no graph to the left of $a$ or to the right of $b$ inside the interval.

If a function has a hole, a jump, or an infinite break anywhere inside the interval, then it is not continuous on that interval. If it has a break at an endpoint, it may still be continuous on an open interval like $(a,b)$, depending on the context.

For AP Calculus AB, this distinction is useful when using the Intermediate Value Theorem, because that theorem requires continuity on a closed interval $[a,b]$.

How to identify the type of discontinuity

When students faces a graph, table, or formula, use a step-by-step process:

1. Check whether $f(c)$ exists.
2. Compute or estimate $\lim_{x\to c^-} f(x)$.
3. Compute or estimate $\lim_{x\to c^+} f(x)$.
4. Compare the one-sided limits.
5. Decide whether the issue is a hole, jump, or infinite behavior.

Here is a quick guide:

• If the limit exists but $f(c)$ is missing or different, it is a removable discontinuity.
• If both one-sided limits exist but are unequal, it is a jump discontinuity.
• If values grow without bound near $c$, it is an infinite discontinuity.

Example: Suppose a graph approaches the same height from both sides at $x=4$, but there is an open circle there and no filled point. That is likely removable.

Example: Suppose a graph approaches $2$ from the left and $5$ from the right at $x=-1$. That is a jump.

Example: Suppose a graph shoots upward near $x=0$ and has a vertical asymptote. That is infinite.

Why discontinuities matter in calculus

Discontinuities are not just graph features; they affect how calculus works. Many theorems in AP Calculus AB depend on continuity.

For instance, the Intermediate Value Theorem says that if a function is continuous on $[a,b]$, then it takes every value between $f(a)$ and $f(b)$. If the function has a discontinuity, the theorem may fail.

Example: A function might start at $f(a)=1$ and end at $f(b)=5$, but if there is a jump in the middle, it may never take the value $3$. Continuity is what guarantees the “no skipping” property.

Discontinuities also affect derivatives. A function with a discontinuity at a point is not differentiable there, because differentiability requires continuity. So learning discontinuities helps students understand later topics like derivatives, chain rule behavior, and motion problems.

In real life, continuity can represent smooth change, while discontinuity can represent sudden change, missing data, or a system switch. Calculus uses these ideas to describe and analyze both kinds of behavior.

Conclusion

Exploring types of discontinuities helps students understand exactly how and why a function can fail to be continuous. A removable discontinuity is a hole that can often be fixed, a jump discontinuity happens when left and right behavior disagree, and an infinite discontinuity happens when the function grows without bound near a point. These ideas connect directly to limits, continuity on intervals, vertical asymptotes, and important theorems like the Intermediate Value Theorem.

In AP Calculus AB, being able to identify and explain discontinuities is a core skill. When you analyze a graph or formula, always ask what happens to the limit, what happens at the point, and whether the function can be made continuous. That habit will help you throughout the course 📘.

Study Notes

• Continuity at $x=c$ requires all three conditions: $f(c)$ is defined, $\lim_{x\to c} f(x)$ exists, and $\lim_{x\to c} f(x)=f(c)$.
• A removable discontinuity happens when the limit exists but the function is missing the point or has the wrong value there.
• A jump discontinuity happens when $\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x)$.
• An infinite discontinuity happens when $f(x)$ grows without bound near a point, often creating a vertical asymptote.
• To classify a discontinuity, compare $f(c)$, $\lim_{x\to c^-} f(x)$, and $\lim_{x\to c^+} f(x)$.
• A function continuous on $[a,b]$ must be continuous at every point in the interval, with one-sided limits used at endpoints.
• Discontinuities matter because theorems like the Intermediate Value Theorem require continuity.
• If a function is discontinuous at a point, it is not differentiable at that point.
• Common AP Calculus AB tasks include reading graphs, simplifying formulas, and using limits to identify the type of discontinuity.