1.10 Exploring Types of Discontinuities

A discontinuity is any point where a function is not continuous, so the graph has a hole, jump, or vertical asymptote. The three types you need to recognize are removable discontinuities, jump discontinuities, and discontinuities from vertical asymptotes. For AP Calculus, use one-sided limits and the function value to classify the discontinuity.

Why This Matters for the AP Calculus Exam

Recognizing and classifying discontinuities is a core skill in AP Calculus, and it shows up across graphical, numerical, and analytic problems. Once you can name the type of discontinuity, you can justify whether a limit exists, decide if a function is continuous at a point, and explain why using one-sided limits.

This topic connects directly to nearby skills like defining continuity at a point and removing discontinuities. Multiple-choice questions often show a graph, table, or piecewise function and ask you to identify the type of discontinuity or to determine whether a limit exists there. Being precise with limit notation and one-sided limits is important for clear exam work.

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Key Takeaways

• A function is continuous at $x = c$ when $f(c)$ exists, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$.
• Removable discontinuity (hole): the two-sided limit exists, but $f(c)$ either does not exist or does not equal that limit.
• Jump discontinuity: the left-hand limit and right-hand limit both exist but are not equal, so the two-sided limit does not exist.
• Vertical asymptote (infinite) discontinuity: the function values grow without bound, so the limit is $+\infty$ or $-\infty$.
• Use one-sided limits to tell jump and asymptote discontinuities apart, and check the function value to spot a removable hole.
• A removable discontinuity from a rational function usually comes from a common factor that cancels in the numerator and denominator.

The Three Types of Discontinuities

To see what a discontinuity is, start with continuity. A function is continuous at a point when all three of these hold:

1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists (the limit from both sides agrees).
3. $\lim_{x \to c} f(x) = f(c)$.

If any one of these fails, the function is discontinuous at that point. The quick mental picture: if you have to lift your pencil to keep drawing the graph, there is a discontinuity there.

Removable Discontinuities (Holes)

A removable discontinuity happens at a single point, often called a hole. The two-sided limit at that point exists, but the function either is not defined there or is defined as a different value. On a graph, you draw the curve as usual but place an open circle at the missing point.

There are two common sources:

• A "blip" in a piecewise function, where one input is defined separately and lands off the curve.
• A common factor in a rational function that cancels.

Example: A Piecewise Hole

Try drawing this piecewise function:

$y = \begin{cases} x^2 & \text{if } x < 1 \\ x - 1 & \text{if } x = 1 \\ -x + 2 & \text{if } x > 1 \end{cases}$

As $x$ approaches 1 from both sides, the curve heads toward the same value, so $\lim_{x \to 1} f(x)$ exists. But $f(1) = 1 - 1 = 0$, which sits off the curve. Because the function value does not match the limit, $x = 1$ is a removable discontinuity. You have to lift your pencil to place that lone point.

Example: A Common Factor Hole

Now look at a rational function where a factor cancels:

$f(x) = \frac{x^3 - 3x^2 + 2x}{x - 1}$

Factor the numerator first: $x^3 - 3x^2 + 2x = x(x - 1)(x - 2)$. The $(x - 1)$ cancels with the denominator, leaving $y = x(x - 2) = x^2 - 2x$ everywhere except $x = 1$.

If you plug $x = 1$ into the original function, you get $\tfrac{0}{0}$, which is undefined. So the graph follows $x^2 - 2x$ but has a hole at $x = 1$. Because the limit exists there, the discontinuity is removable.

Jump Discontinuities

A jump discontinuity is a vertical "jump" between the two sides of the graph, like the curve shifts up or down suddenly. This happens when the left-hand limit does not equal the right-hand limit. Since the one-sided limits disagree, the two-sided limit does not exist.

Example of a Jump Discontinuity

Graph this piecewise function:

$y = \begin{cases} x - 2 & \text{if } x \leq 3 \\ x + 1 & \text{if } x > 3 \end{cases}$

As $x \to 3^-$, the function approaches $3 - 2 = 1$. As $x \to 3^+$, it approaches $3 + 1 = 4$. The left and right limits are different, so there is a jump of 3 units at $x = 3$.

One circle is filled in and the other is open. The left piece uses $\leq$, so $x = 3$ belongs to it and gets a closed circle. The right piece uses just $>$, so its endpoint is an open circle.

Vertical Asymptote (Infinite) Discontinuities

This discontinuity happens when the function values grow without bound near a point, so the one-sided limits head to $+\infty$ or $-\infty$. The two sides can go in opposite directions or the same direction.

Example of an Asymptote Discontinuity

Graph the simple function $y = \dfrac{1}{x}$.

As $x \to 0^+$, $\tfrac{1}{x} \to +\infty$, and as $x \to 0^-$, $\tfrac{1}{x} \to -\infty$. The function blows up near $x = 0$, giving a vertical asymptote there. Other common examples with vertical asymptotes include $\tan x$ and $\ln x$ near $x = 0$.

How to Use This on the AP Calculus Exam

MCQ

When a problem hands you a graph, table, or piecewise function, work through this checklist:

• Find the left-hand limit and the right-hand limit at the point in question.
• If both one-sided limits exist and agree, the two-sided limit exists. Then compare it to $f(c)$: if they differ or $f(c)$ is undefined, it is a removable discontinuity.
• If the one-sided limits exist but disagree, it is a jump discontinuity.
• If the values go to $\pm\infty$, it is a vertical asymptote discontinuity.

Problem Solving

For rational functions, factor the numerator and denominator before deciding. A factor that cancels gives a removable hole. A factor that stays only in the denominator gives a vertical asymptote. This single step prevents most mix-ups between holes and asymptotes.

Common Trap

Watch the inequality symbols in piecewise functions. The piece that uses $\leq$ or $\geq$ owns the boundary point and gets the closed circle. Getting this wrong can flip your answer about whether the function value matches the limit.

Common Misconceptions

• "A hole means the limit does not exist." The opposite is true for a removable discontinuity: the two-sided limit exists, but the function value is missing or different.
• "A jump discontinuity can still have a two-sided limit." It cannot. The left and right limits disagree, so the two-sided limit does not exist.
• "Any time you get $\tfrac{0}{0}$ there is an asymptote." A $\tfrac{0}{0}$ form often signals a removable hole from a canceling factor, not a vertical asymptote. Factor first to find out.
• "If a factor cancels, the discontinuity disappears completely." The simplified expression matches the original everywhere except the canceled point, where a hole remains unless the function is redefined there.
• "Continuous just means the function is defined at the point." Continuity needs three things: $f(c)$ exists, the limit exists, and they are equal.

Related AP Calculus Guides

• 1.2 Defining Limits and Using Limit Notation
• 1.1 Introducing Calculus: Can Change Occur at An Instant?
• 1.6 Determining Limits Using Algebraic Manipulation
• 1.3 Estimating Limit Values from Graphs
• 1.5 Determining Limits Using Algebraic Properties of Limits
• 1.8 Determining Limits Using the Squeeze Theorem