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In calculus, analysing where and why a function breaks is essential before investigating differentiability and integration. While some functions have missing points (removable discontinuities) or vertical asymptotes (infinite discontinuities), others experience an abrupt, finite break in their graph. This sudden vertical shift between two sections of a curve is known as a jump discontinuity (or step discontinuity).

Jump discontinuities are especially prominent when analysing piecewise-defined functions, absolute value functions (signum functions), and greatest integer (floor/ceiling) functions. In A-Level Mathematics, Further Mathematics, and calculus exams, questions frequently test your ability to evaluate one-sided limits, classify discontinuities, calculate the jump size, and solve for algebraic constants that make piecewise boundaries smooth and continuous.

Explanation for what a jump discontinuty looks like and how it is defined for A Level Maths — Image Source: Gianpiero Placidi

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Core Theory

A function f(x) has a jump discontinuity at x = c if both one-sided limits exist as finite real numbers, but they do not equal each other.

Formal Mathematical Definition:

A function f(x) has a jump discontinuity at x = c if and only if:

\lim_{x \to c^-} f(x) = L_1 \quad (L_1 \in \mathbb{R})

and

\lim_{x \to c^+} f(x) = L_2 \quad (L_2 \in \mathbb{R})

and

L_1 \neq L_2

Because the left-hand limit ( $L_1$ ) and right-hand limit ( $L_2$ ) are not equal, the two-sided limit does not exist (DNE):

The Magnitude (Size) of a Jump

The jump size (or jump magnitude) measures the vertical distance between the two one-sided limit values at x = c:

\text{Jump Size} = \left\vert{} \lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x) \right\vert{} = \vert{}L_2 - L_1\vert{}

If $L_2 > L_1$ , the function undergoes a positive (upward) step from left to right.
If $L_2 < L_1$ , the function undergoes a negative (downward) step from left to right.
If $\vert{}L_2 - L_1\vert{} = 0$ , the left and right limits agree, meaning there is no jump.

Left-Continuity and Right-Continuity

At a jump discontinuity x = c, the function value f(c) might equal one of the one-sided limits, or it might be completely undefined:

Continuous from the Left (Left-Continuous):

\lim_{x \to c^-} f(x) = f(c)

Continuous from the Right (Right-Continuous):

\lim_{x \to c^+} f(x) = f(c)

Even if a function is continuous from one side, the function as a whole remains discontinuous at x = c because the two-sided limit does not exist.

Common Functions with Jump Discontinuities

A. Piecewise Linear and Non-Linear Functions

When different sub-functions are defined over adjacent intervals, the junction points (transition boundaries) often produce jumps:

f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}

B. The Signum / Absolute Value Quotient

The standard absolute value signum function produces a step jump across x = 0:

f(x) = \dfrac{\vert{}x\vert{}}{x} = \begin{cases} -1 & \text{if } x < 0 \\ +1 & \text{if } x > 0 \end{cases}

Here, $\lim_{x \to 0^-} f(x) = -1$ and $\lim_{x \to 0^+} f(x) = +1$ . The jump size is $\vert{}1 - (-1)\vert{} = 2$ .

C. The Greatest Integer (Floor) Function

The floor function $f(x) = \lfloor x \rfloor$ rounds $x$ down to the nearest integer, generating an infinite series of jump discontinuities at every integer $x = n \in \mathbb{Z}$ :

\lim_{x \to n^-} \lfloor x \rfloor = n - 1, \quad \lim_{x \to n^+} \lfloor x \rfloor = n

Worked Example

Confirming and Classifying a Jump Discontinuity

A piecewise function f(x) is defined by:

f(x) = \begin{cases} 3x - 1 & \text{if } x < 2 \\ x^2 + 3 & \text{if } x \ge 2 \end{cases}

Determine whether f(x) has a jump discontinuity at x = 2. If so, calculate the size of the jump.

Step 1: Evaluate the left-hand limit ( $x \to 2^-$ ).

\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (3x - 1) = 3(2) - 1 = 5

Step 2: Evaluate the right-hand limit ( $x \to 2^+$ ).

\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x^2 + 3) = (2)^2 + 3 = 7

Step 3: Compare limits and classify.

Since $\lim_{x \to 2^-} f(x) = 5$ and $\lim_{x \to 2^+} f(x) = 7$ are both finite but unequal:

\lim_{x \to 2} f(x) = \text{DNE}

Therefore, f(x) has a jump discontinuity at x = 2.

Step 4: Calculate the jump size.

\text{Jump Size} = \vert{}7 - 5\vert{} = 2

Practice Questions & Solutions

Score:

A piecewise function f(x) is defined by:

$\text{[math]}$

Evaluate the one-sided limits

$\text{[math]}$

Explain why f(x) has a jump discontinuity at x = 3.

Calculate the magnitude (size) of the jump.

Solution

Step 1: Evaluate one-sided limits.

$\text{[math]}$

Step 2: Classify the discontinuity.

Both one-sided limits exist as finite real numbers, but they are not equal $\text{[math]}$

Therefore, the two-sided limit $\text{[math]}$ Does Not Exist, creating a jump discontinuity at x = 3

Step 3: Calculate the jump size.

$\text{[math]}$

A function g(x) is defined on $\text{[math]}$ by:

$\text{[math]}$

Where k is a constant.

Determine the value of k that eliminates the jump discontinuity, making g(x) continuous at x = 2

State the value of the two-sided limit $\text{[math]}$ for this value of k

Solution

Step 1: Set the condition for continuity (no jump).

To prevent a jump discontinuity at the boundary x = 2, the left-hand limit and right-hand limit must be equal:

$\text{[math]}$

Step 2: Evaluate both one-sided limits in terms of k.

$\text{[math]}$

Step 3: Solve for k.

$\text{[math]}$

Step 4: State the resulting two-sided limit.

Substituting k = 5 into either branch:

$\text{[math]}$

Consider the function:

$\text{[math]}$

Evaluate the left-hand limit:

$\text{[math]}$

and the right-hand limit:

$\text{[math]}$

State the nature of the discontinuity at x = 4 and determine the jump size

Solution

Step 1: Apply the definition of absolute value.

$\text{[math]}$

Step 2: Evaluate the left-hand limit ( $\text{[math]}$ ).

$\text{[math]}$

Step 3: Evaluate the right-hand limit ( $\text{[math]}$ ).

$\text{[math]}$

Step 4: Classification & Jump Size.

Because the one-sided limits are finite and unequal ( $\text{[math]}$ ), h(x) has a jump discontinuity at x = 4.

$\text{[math]}$

A function p(x) is defined by:

$\text{[math]}$

Where a is a real constant.

Evaluate:

$\text{[math]}$

Find the value of a for which there is no jump discontinuity at x = 0

If a = 1, calculate the jump size at x = 0

Solution

Step 1:

Using the standard identity

$\text{[math]}$ :

$\text{[math]}$

Step 2: Find a to remove the jump.

Evaluate the right-hand limit:

$\text{[math]}$

Equating the one-sided limits:

$\text{[math]}$

Step 3: Jump size when a = 1.

Left-hand limit: $\text{[math]}$

Right-hand limit: $\text{[math]}$

$\text{[math]}$

A piecewise function q(x) is defined on all real numbers by:

$\text{[math]}$

Find the values of constants a and b such that q(x) has no jump discontinuities on $\text{[math]}$

Solution

For q(x) to be continuous across $\text{[math]}$ , no jump discontinuities can occur at the boundary points x = 1 and x = 4.

Step 1: Continuity at x = 1.

$\text{[math]}$

Step 2: Continuity at x = 4.

$\text{[math]}$

Step 3: Solve for a.

Substitute b = -9 into Equation 1:

$\text{[math]}$

Final Values: $\text{[math]}$

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

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what will be sign of -infinity is to power infinity plus or minus

Vanessa - Editorial Manager Superprof UK

July 2026

The sign **cannot be determined**.

For a negative base:

* **Even exponent** → **positive**
* **Odd exponent** → **negative**

So as the exponent tends to infinity, the sign depends on whether the exponent is even or odd. Therefore, ((-\infty)^\infty) has **no definite sign**—it can be **positive or negative**.

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Chapters

Core Theory
Worked Example
Practice Questions & Solutions

What is a Jump Discontinuity: Examples of Jump Discontinuity

Core Theory

The Magnitude (Size) of a Jump

Left-Continuity and Right-Continuity

Common Functions with Jump Discontinuities

A. Piecewise Linear and Non-Linear Functions

B. The Signum / Absolute Value Quotient

C. The Greatest Integer (Floor) Function

Worked Example

Confirming and Classifying a Jump Discontinuity

Practice Questions & Solutions

Theory

Calculating Limits

Continuity

Continuity on a Closed Interval

Essential Discontinuity

One-Sided Limit

Bolzano’s Theorem

Limits at Infinity

Infinite Limit

Limit of a Logarithmic Function

Removable Discontinuity

Limit of an Exponential Function

Limit Rules

The Properties of Infinity in Calculus

One to the Power of Infinity

Limits and Dividing by Zero: Theory and Examples

Understanding All the Indeterminate Forms

Zero Over Zero: Indeterminate Forms and L’Hôpital’s Rule

What Is Zero Times Infinity? Theory and Practice Problems

Infinity Minus Infinity: Worked Examples and Solutions

Extreme Value Theorem in Calculus

Intermediate Value Theorem: Theory and Worked Examples

Discontinuous Function Graph Examples And Problems

What is a Continuous Function?

Jump Discontinuity Example Questions With Full Solutions

Formulas

Continuity Formulas

Limit Formulas

Exercises

Limit Problems

Intermediate Value Theorem Problems

Continuity Worksheet

Continuity Practice Problems for A Level Maths

Limits Worksheet With Worked Solutions

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