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In calculus, understanding the continuity of a function is fundamental before moving on to differentiation and integration. Intuitively, a function is continuous if you can sketch its curve without lifting your pen from the paper. When a graph contains a break, hole, vertical asymptote, or jump, the function is described as discontinuous.

Evaluating limits at points of discontinuity is a core topic across A-Level Mathematics, Further Mathematics, and introductory university calculus courses. Exam questions frequently require you to classify discontinuities, evaluate one-sided limits, and determine unknown constants to ensure a piecewise function is continuous.

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The Three Formal Conditions for Continuity

A function f(x) is continuous at a specific point x = c if and only if all three of the following conditions are met:

The 3 Conditions for Continuity at x = c:

f(c) is defined (c lies within the domain of f).
$\lim_{x \to c} f(x)$ exists (the left-hand limit equals the right-hand limit: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$ ).
$\lim_{x \to c} f(x) = f(c)$ (the value of the limit matches the actual output of the function).

If any of these three conditions fails, f(x) is discontinuous at x = c.

The Three Main Types of Discontinuity

Discontinuities are classified by the behavior of the function's one-sided limits and functional values at the break point.

Graphs depicting three types of discontinuity in functions: removable, jump and infinite — Image Source: Gianpiero Placidi

Removable Discontinuity (Point / Hole)

A removable discontinuity occurs when the two-sided limit exists, but does not equal the function's value (or the function is undefined at that point).

Limit Condition: $\lim_{x \to c} f(x) = L$ (exists and is finite).
Function Condition: Either f(c) is undefined, or $f(c) \neq L$ .
Why it's called "removable": The discontinuity can be removed by redefining or filling in a single point $f(c) = L$ .
Common occurrence: Rational functions where numerator and denominator share a common factor, such as:

f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \quad (x \neq 2)

Jump Discontinuity (Step Discontinuity)

A jump discontinuity occurs when the curve "jumps" abruptly from one value to another.

Limit Condition: Both one-sided limits exist as finite numbers, but they are not equal:

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

Two-sided Limit: $\lim_{x \to c} f(x)$ does not exist (DNE).
Common occurrence: Piecewise functions, floor/ceiling functions ( $\lfloor x \rfloor$ ), and signum/absolute-value quotients like:

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

Infinite / Essential Discontinuity (Asymptotic)

An infinite discontinuity (often referred to more broadly as an essential discontinuity) occurs when the function approaches positive or negative infinity as x approaches c from either side.

Limit Condition: At least one of the one-sided limits is unbounded:

\lim_{x \to c^-} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to c^+} f(x) = \pm\infty

Graphical Feature: A vertical asymptote at x = c.
Common occurrence: Rational functions where the denominator equals zero while the numerator is non-zero, such as:

f(x) = \frac{1}{x - 3} \quad \text{or} \quad f(x) = \frac{1}{(x - 2)^2}

Worked Example

Evaluate the limit and classify the discontinuity of:

f(x) = \frac{x^2 - 9}{x - 3} \quad \text{at } x = 3

Step 1: Test direct substitution:

f(3) = \frac{3^2 - 9}{3 - 3} = \frac{0}{0} \quad (\text{Indeterminate form})

$f(3)$ is undefined.

Step 2: Factor the numerator and evaluate the limit:

\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 3 + 3 = 6

Step 3: Classify the discontinuity:

Because $\lim_{x \to 3} f(x) = 6$ exists as a finite value, but $f(3)$ is undefined, $f(x)$ has a removable discontinuity at $x = 3$ .

Practice Questions & Solutions

Score:

A function f(x) is defined as:

$\text{[math]}$

Identify all values of x where f(x) is discontinuous.

For each point of discontinuity, classify whether it is removable or infinite.

Solution

Find points of discontinuity (where denominator equals zero):

$\text{[math]}$

Factor the numerator and denominator:

$\text{[math]}$

Analyze x = 3:

$\text{[math]}$

Because the limit exists as a finite real number, x = 3 is a removable discontinuity.

Analyze x = -3:

As $\text{[math]}$ , the numerator approaches (-3 + 2) = -1, while the denominator $\text{[math]}$

$\text{[math]}$

Because the limit is unbounded, x = -3 is an infinite discontinuity (vertical asymptote at x = -3).

For the function:

$\text{[math]}$

determine the point where the function is discontinuous, evaluate the limit as x approaches this point to classify the type of discontinuity, and state the value that should be assigned to g(x) at this point to make the function continuous.

Solution

Step 1: Identify the point of discontinuity.

Setting the denominator to zero gives $\text{[math]}$

The function is discontinuous at x = 2 because g(2) is undefined.

Step 2: Factor and evaluate the limit as $\text{[math]}$

$\text{[math]}$

Step 3: Classify the discontinuity.

Because the two-sided limit exists and is finite:

$\text{[math]}$

the graph exhibits a removable discontinuity (a point / hole) at x = 2.

Step 4: Define g(2) for continuity.

To satisfy the continuity condition:

$\text{[math]}$

define g(2) = -1.

Evaluate the limit:

$\text{[math]}$

and state how f(1) should be defined to make $\text{[math]}$ continuous at x = 1.

Solution

Factor difference of cubes:

$\text{[math]}$

Evaluate limit:

$\text{[math]}$

Condition for continuity:

For f(x) to be continuous at x = 1, we must define $\text{[math]}$

Therefore, set f(1) = 3

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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.

Formulas

Continuity Formulas

Limit Formulas

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

The Three Formal Conditions for Continuity
The Three Main Types of Discontinuity
Worked Example
Practice Questions & Solutions

Discontinuous Function Examples: What is Function Discontinuity?

The Three Formal Conditions for Continuity