Types Of Discontinuity: Removable, Jump & Infinite

What Are The Types Of Discontinuity?

The types of discontinuity are the distinct ways a function can fail to be continuous at a single point. Before naming them, fix what continuity means. A function $f$ is continuous at a point $a$ when three things all hold and agree:

$$f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)$$

Unpacked, that one line hides three separate conditions:

• $f(a)$ is defined (the point exists),

• $\displaystyle\lim_{x \to a} f(x)$ exists (both one-sided limits agree on a finite value),

• the two are equal.

A discontinuity at $a$ is the failure of any one of these. Which condition fails decides the type. There are three standard types plus one that is genuinely wilder:

• Removable: the limit exists, but it does not equal $f(a)$ (or $f(a)$ is missing).

• Jump: the one-sided limits are both finite but disagree, so the two-sided limit fails.

• Infinite: at least one one-sided limit is $\pm\infty$, so the curve runs up a vertical asymptote.

• Essential (oscillating): the function oscillates so fast that no one-sided limit exists at all.

The first three are the set every syllabus tests. The fourth is worth knowing because it is the honest answer to "what if even the one-sided limits do not exist." For the underlying machinery, see limit of a function and continuity of a function.

What Is A Removable Discontinuity?

A removable discontinuity is a single hole in an otherwise smooth curve. The two-sided limit exists and is finite, but the function either is undefined there or is defined at the wrong height.

$$\lim_{x \to a} f(x) = L \text{ (finite)}, \qquad \text{but } f(a) \neq L \text{ or } f(a) \text{ is undefined.}$$

It earns the name "removable" because you can patch it: redefine $f(a) = L$ and the function becomes continuous, no other change needed.

Example 1: A cancelled factor.

Classify the discontinuity of $f(x) = \dfrac{x^2 - 1}{x - 1}$ at $x = 1$.

At $x = 1$ the formula gives $\dfrac{0}{0}$, so $f(1)$ is undefined. Factor and cancel for every $x \neq 1$:

$$f(x) = \frac{(x-1)(x+1)}{x-1} = x + 1, \qquad x \neq 1.$$

The limit is therefore finite:

$$\lim_{x \to 1} f(x) = \lim_{x \to 1}(x+1) = 2.$$

The limit exists ($2$) but $f(1)$ does not, so this is a removable discontinuity. Patching it with $f(1) = 2$ removes the hole.

Final answer: removable discontinuity at $x = 1$; fill it by setting $f(1) = 2$.

Geometrically, the graph is the straight line $y = x + 1$ with a single open circle punched out at the point $(1, 2)$. For the dedicated deep dive, see removable discontinuity.

What Is A Jump Discontinuity?

A jump discontinuity happens when the left-hand and right-hand limits both exist and are finite, but they are not equal. The curve arrives at one height from the left and leaves from a different height on the right, so it jumps.

$$\lim_{x \to a^-} f(x) = L_1, \quad \lim_{x \to a^+} f(x) = L_2, \qquad L_1 \neq L_2 \text{ (both finite).}$$

Because the two sides disagree, the two-sided limit does not exist, so no patch at a single point can fix it. The size of the jump is $\lvert L_2 - L_1 \rvert$.

Example 2: The sign function.

Classify the discontinuity of $f(x) = \dfrac{\lvert x \rvert}{x}$ at $x = 0$.

For $x > 0$, $\lvert x \rvert = x$, so $f(x) = 1$. For $x < 0$, $\lvert x \rvert = -x$, so $f(x) = -1$. The one-sided limits are:

$$\lim_{x \to 0^-} f(x) = -1, \qquad \lim_{x \to 0^+} f(x) = 1.$$

Both are finite and they disagree, so this is a jump discontinuity. The jump has size $\lvert 1 - (-1) \rvert = 2$.

Final answer: jump discontinuity at $x = 0$, with a jump of $2$.

Geometrically the graph sits flat at height $-1$ for negative $x$ and flat at height $+1$ for positive $x$, with a clean vertical gap of $2$ units at the origin. Piecewise-defined functions are the usual home of jumps; see piecewise functions and the focused page on the jump discontinuity.

What Is An Infinite Discontinuity?

An infinite discontinuity occurs when at least one one-sided limit is infinite. The function grows without bound near $a$, and the graph hugs a vertical asymptote it never touches.

$$\lim_{x \to a^-} f(x) = \pm\infty \quad \text{and/or} \quad \lim_{x \to a^+} f(x) = \pm\infty.$$

Example 3: A simple reciprocal.

Classify the discontinuity of $f(x) = \dfrac{1}{x - 2}$ at $x = 2$.

As $x$ approaches $2$ from the right, $x - 2$ is a tiny positive number, so the quotient grows huge and positive. From the left, $x - 2$ is a tiny negative number, so the quotient dives huge and negative:

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

Both one-sided limits are infinite, so this is an infinite discontinuity, and the line $x = 2$ is a vertical asymptote.

Final answer: infinite discontinuity at $x = 2$; vertical asymptote $x = 2$.

Geometrically the curve rockets up toward $+\infty$ just right of $x = 2$ and plunges toward $-\infty$ just left of it. The deep dive lives at infinite discontinuity.

Is There An Oscillating (Essential) Discontinuity?

Yes. The three types above are sometimes called the "main" types because each one-sided limit at least exists (finite for removable and jump, infinite for infinite). An essential or oscillating discontinuity is the case where a one-sided limit does not exist in any form, because the function oscillates infinitely often near the point.

The standard example is $f(x) = \sin\left(\dfrac{1}{x}\right)$ at $x = 0$. As $x \to 0$, the input $\dfrac{1}{x}$ races off to infinity, so $\sin(1/x)$ sweeps between $-1$ and $1$ endlessly, never settling.

$$\lim_{x \to 0} \sin\left(\frac{1}{x}\right) \text{ does not exist (no finite value, and not } \pm\infty).$$

There is no honest way to assign it a limit, and no single redefinition helps, which is exactly why it is called essential rather than removable. It is the reader's signal that "the limit does not exist" and "the function blows up" are not the same failure.

How Do You Classify A Discontinuity?

Every classification comes down to four quantities at the point $a$: the left limit, the right limit, the two-sided limit, and the value $f(a)$. Compute the one-sided limits first, and the type falls out.

Table: The one-sided-limit test for the types of discontinuity.

Read the table left to right. If both one-sided limits are finite and equal, you are in the removable row (a hole), and the only remaining question is whether $f(a)$ matches. If they are finite but differ, it is a jump; if either runs to $\pm\infty$, it is infinite. If a one-sided limit refuses to exist at all, it is essential.

This single test replaces guessing from a graph. To build the underlying skill, revisit limits and derivatives.

Example 4: One function, two different discontinuities.

Classify every discontinuity of $f(x) = \dfrac{x - 5}{x^2 - 25}$.

Factor the denominator: $x^2 - 25 = (x-5)(x+5)$, so $f$ is undefined at $x = 5$ and $x = -5$. For $x \neq 5$,

$$f(x) = \frac{x-5}{(x-5)(x+5)} = \frac{1}{x+5}.$$

At $x = 5$ the factor cancelled, so the limit is finite:

$$\lim_{x \to 5} f(x) = \frac{1}{5+5} = \frac{1}{10} = 0.1000.$$

Finite limit, undefined value, so $x = 5$ is a removable discontinuity. At $x = -5$ nothing cancels, and $\dfrac{1}{x+5}$ runs to $\pm\infty$, so $x = -5$ is an infinite discontinuity (vertical asymptote).

Final answer: removable at $x = 5$ (limit $0.1000$), infinite at $x = -5$.

Why Do The Types Of Discontinuity Matter?

Classifying a break is not busywork. The type tells you what, if anything, you are allowed to do next.

• A removable hole is safe to repair. Because the limit exists, the function is "one point away" from continuous. Every simplification you do in calculus, cancelling a common factor, evaluating a $0/0$ limit, quietly relies on the discontinuity being removable rather than real.

• A jump cannot be smoothed. No redefinition at the point closes a gap between two different one-sided limits, so a jump signals a genuine switch in behaviour, not a typo in the formula.

• An infinite discontinuity marks a boundary. The vertical asymptote is a wall the function cannot cross, which is why an integral straddling one is improper and a naive evaluation across it is invalid.

• Continuity is the entry ticket for the big theorems. The Intermediate Value Theorem and the Extreme Value Theorem both assume continuity on a closed interval. A single jump or infinite break voids them, so knowing the type tells you which theorems still apply.

Seen this way, the classification is really a diagnosis: removable means "curable," jump means "a real step," infinite means "a wall," and essential means "no limit lives here at all." For how continuity feeds differentiability, see continuity and differentiability.

Who Shaped The Modern Idea Of Continuity?

For a century after Newton and Leibniz, "continuous" meant little more than "a curve you can draw without lifting your pen." Turning that picture into the precise limit conditions above took three careful thinkers.

Two more names finished the job of making continuity exact:

• Augustin-Louis Cauchy (1789–1857, France) put limits at the centre of calculus in his 1820s lecture courses, defining continuity through the behaviour of $f(x) - f(a)$ as $x \to a$.

• Karl Weierstrass (1815–1897, Germany) supplied the airtight $\varepsilon$–$\delta$ language still used today, and famously exhibited a function that is continuous everywhere yet differentiable nowhere, proving that continuity is a genuinely separate idea from smoothness.

Where Are The Types Of Discontinuity Used In The Real World?

Breaks in a graph are not just exam objects. Real systems change abruptly, and the type of discontinuity names what kind of change it is.

• Signal processing: a dropped audio sample or a lost video frame is a jump discontinuity in the signal, and engineers design filters specifically to detect and smooth these gaps.

• Physics and phase changes: at its boiling point, water's density falls in a sudden jump as liquid becomes vapour, a discontinuity in an otherwise continuous relationship between temperature and state.

• Economics and tax systems: a tax bracket or a shipping-cost threshold creates a jump, where earning one extra unit pushes the whole calculation to a new rule.

• Control systems and electronics: a circuit driven past a limit can send a voltage toward a rail, an infinite-style blow-up that hardware must be protected against.

• Computer graphics: a hole or crack in a rendered surface is a removable discontinuity in the underlying function, patched by redefining the value at the missing point.

One vocabulary, hole versus jump versus wall, lets a physicist, an economist, and an engineer describe "where the smooth behaviour breaks" in the same precise language.

What Are The Most Common Mistakes With Types Of Discontinuity?

These four errors account for most lost marks, and each matches a question real students ask on r/calculus, askIITians, and AP review guides.

Assuming every hole is removable.

Where it slips in:

A student sees the function is undefined at a point and labels it removable without checking whether a finite limit actually exists.

Don't do this:

Do not equate "undefined" with "removable." Removable requires the two-sided limit to exist and be finite.

The correct way:

Compute the one-sided limits first. Only if they agree on a finite value is the discontinuity removable; if they run to $\pm\infty$, it is infinite instead.

Confusing "limit does not exist" with an infinite discontinuity.

Where it slips in:

A student meets a jump, sees that the two-sided limit fails, and writes "infinite discontinuity" because the limit "does not exist."

Don't do this:

Do not treat every missing two-sided limit as infinite. A jump also kills the two-sided limit, yet both one-sided limits are finite.

The correct way:

Check whether the one-sided limits are finite (jump) or actually reach $\pm\infty$ (infinite). The word "infinite" refers to the limit's value, not merely to its absence.

Forgetting to compare the limit with $f(a)$.

Where it slips in:

A student finds that $\lim_{x \to a} f(x)$ exists and stops, declaring the function continuous without checking the actual value at $a$.

Don't do this:

Do not skip the third continuity condition. A finite limit is necessary but not sufficient.

The correct way:

Confirm $\lim_{x \to a} f(x) = f(a)$. If the limit exists but $f(a)$ is missing or different, the point is a removable discontinuity, not a continuous point.

Calling every undefined point a vertical asymptote.

Where it slips in:

A student sees a zero denominator and immediately writes "vertical asymptote," even when a factor cancels.

Don't do this:

Do not judge from the denominator alone. A cancelling factor produces a hole, not a wall.

The correct way:

Simplify first. If the problem factor cancels and the limit is finite, it is a removable hole; only a non-cancelling zero denominator gives an infinite discontinuity.

Practice Problems On Types Of Discontinuity

Classify the discontinuity at the stated point, then check against the verified answer.

1. $f(x) = \dfrac{x^2 - 9}{x - 3}$ at $x = 3$.
   (Answer: $\dfrac{(x-3)(x+3)}{x-3} = x+3$, so $\lim_{x\to3} = 6$; the value is undefined. Removable; patch with $f(3) = 6$.)

2. $f(x) = \begin{cases} 2x, & x < 1 \ x + 3, & x \ge 1 \end{cases}$ at $x = 1$.
   (Answer: $\lim_{x\to1^-} = 2$, $\lim_{x\to1^+} = 4$; both finite, unequal. Jump of size $2$.)

3. $f(x) = \dfrac{1}{x + 1}$ at $x = -1$.
   (Answer: $\lim_{x\to-1^+} = +\infty$, $\lim_{x\to-1^-} = -\infty$. Infinite; vertical asymptote $x = -1$.)

4. $f(x) = \lfloor x \rfloor$ (the greatest-integer function) at $x = 2$.
   (Answer: $\lim_{x\to2^-} = 1$, $\lim_{x\to2^+} = 2$. Jump of size $1$.)

5. $f(x) = \sin\left(\dfrac{1}{x}\right)$ at $x = 0$.
   (Answer: the one-sided limits do not exist; the graph oscillates. Essential / oscillating.)

6. $f(x) = \dfrac{x + 4}{x^2 - 16}$ at $x = 4$ and $x = -4$.
   (Answer: $\dfrac{x+4}{(x-4)(x+4)} = \dfrac{1}{x-4}$; at $x=-4$ limit $= -\dfrac{1}{8} = -0.1250$, removable; at $x = 4$ it runs to $\pm\infty$, infinite.)

Where Can You Get Help With Types Of Discontinuity?

Limits and continuity are the first real hurdle in calculus, and a hole-versus-jump slip early on tends to follow a student into integrals and series. Working a few classified examples aloud with a trainer fixes the one-sided-limit habit faster than re-reading definitions. Bhanzu's programs pair a learner with a live teacher who starts from the graph and builds the test up from there:

• High school math tutoring for AP-level calculus foundations.

• GCSE maths tutor for UK students meeting limits and continuity.

• Online math tutor from India for one-to-one calculus support.

• Bhanzu math classes for a structured start-to-finish path.

Where Should You Go Next After Types Of Discontinuity?

Each type opens onto a deeper page, so pick the door that matches what you are studying.

1. Removable discontinuity. The full method for spotting and patching holes, including $0/0$ limits.

2. Jump discontinuity. One-sided limits, step and piecewise functions, and jump size in detail.

3. Continuity of a function. The three-condition definition these types all break, and the theorems continuity unlocks.

If your child is meeting limits and continuity for the first time, a live Bhanzu trainer teaches the types of discontinuity from the graph up, so the one-sided-limit test becomes second nature, in the Bhanzu math program.