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Whether you are sketching curves, evaluating derivatives, or determining area under a graph, continuity is one of the foundational building blocks of mathematical analysis.

Intuitively, a function is continuous if you can draw its graph on paper without lifting your pencil—meaning there are no gaps, holes, or sudden vertical jumps. However, in A-Level Maths and introductory university calculus, intuition must be backed up by a precise, formal definition using limits.

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What is a Continuous Function?

An informal way to visualise a continuous function is a smooth, uninterrupted curve over its entire domain. If a graph breaks, leaps across a vertical gap, or shoots off to infinity, the function is discontinuous at that specific point.

Graph illustrating how a continuous function has no breaks, gaps or asymptotes — Image Source: Gianpiero Placidi

The Formal 3-Step Test for Continuity at a Point

To prove mathematically that a function $f(x)$ is continuous at a specific point $x = c$ , it must satisfy all three of the following conditions:

\text{1. } f(c) \text{ is defined } (c \text{ is in the domain of } f)

ie. is f(c) a solid point on the graph?

\text{2. } \lim_{x \to c} f(x) \text{ exists } \left(\text{meaning } \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)\right)

ie. does the left-hand limit = right-hand limit?

\text{3. } \lim_{x \to c} f(x) = f(c)

ie. does the limit equal the exact value?

If any single condition fails, the function f(x) is discontinuous at x = c.

Algebra and Properties of Continuous Functions

If two functions $f(x)$ and $g(x)$ are both continuous at $x = c$ , then the following algebraic combinations are also continuous at $x = c$ :

Sum and Difference: $(f \pm g)(x) = f(x) \pm g(x)$

Constant Multiple: $(k \cdot f)(x) = k \cdot f(x)$ for any real constant $k$

Product: $(f \cdot g)(x) = f(x) \cdot g(x)$

Quotient: $\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$ , provided $g(c) \neq 0$

Composite Function: $(f \circ g)(x) = f(g(x))$ , provided $g$ is continuous at $c$ and $f$ is continuous at $g(c)$

Standard Functions Continuous on Their Domains

The following basic family of functions are continuous at every point in their respective domains:

Polynomial functions (e.g., $f(x) = x^3 - 4x + 2$ ) — continuous for all $x \in \mathbb{R}$ .

Rational functions (e.g., $f(x) = \dfrac{P(x)}{Q(x)}$ ) — continuous at all points where $Q(x) \neq 0$ .

Exponential functions (e.g., $f(x) = e^x$ ) — continuous for all $x \in \mathbb{R}$ .

Logarithmic functions (e.g., $f(x) = \ln(x)$ ) — continuous for all $x > 0$ .

Trigonometric functions ( $\sin x$ and $\cos x$ on $\mathbb{R}$ ; $\tan x$ wherever $\cos x \neq 0$ ).

Types of Discontinuity

When a function fails the 3-step continuity test at $x = c$ , the failure generally falls into one of three distinct categories:

1. Removable Discontinuity (Point Discontinuity)

A removable discontinuity occurs when the limit $\lim_{x \to c} f(x)$ exists, but either $f(c)$ is undefined or $f(c) \neq \lim_{x \to c} f(x)$ Graphically, this appears as a single "hole" in the curve:

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L \quad \text{but} \quad f(c) \neq L

2. Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and right-hand limit both exist as finite numbers, but they are not equal. Graphically, the curve "jumps" vertically from one line level to another.

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

3. Infinite Discontinuity (Essential Discontinuity)

An infinite discontinuity occurs when one or both of the one-sided limits approach $\pm \infty$ . Graphically, this represents a vertical asymptote at $x = c$ .

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

Continuity in Piecewise Functions

A piecewise function is defined by different sub-functions across different intervals of its domain. A piecewise function is continuous overall if:

Each constituent sub-function is continuous on its open interval.
The left-hand limit, right-hand limit, and exact function value match at every boundary transition point.

Illustration of a continuous piecewise function meeting all the boundary conditions — Image Source: Gianpiero Placidi

Are Radical Functions Continuous?

Radical functions of the form:

f(x) = \sqrt[n]{x}

are continuous at every point across their entire domain. For odd root indices (such as the cube root $f(x) = \sqrt[3]{x}$ ), the domain spans all real numbers ( $-\infty, \infty$ ), making the function continuous everywhere on $\mathbb{R}$ . For even root indices (such as the standard square root $f(x) = \sqrt{x}$ ), the function is defined and continuous on the interval $[0, \infty)$ , satisfying right-continuity at the boundary endpoint $x = 0$ because:

\lim_{x \to 0^+} \sqrt{x} = f(0) = 0

In exam questions, note that values outside the domain (such as negative inputs for even roots) are points where the function is undefined, rather than points of discontinuity—a function is considered continuous if it is continuous at every point where it is defined.

Worked Examples & Solutions

Example 1: Testing Continuity at a Point

Question: Test whether the function $f(x) = \dfrac{x^2 - 4}{x - 2}$ is continuous at $x = 2$ . If discontinuous, identify the type of discontinuity.

Step-by-Step Solution:

Check Condition 1 ( $f(2)$ exists):

f(2) = \dfrac{2^2 - 4}{2 - 2} = \dfrac{0}{0} \quad \text{(Undefined)}

Because $f(2)$ is undefined, $f(x)$ fails Condition 1 and is discontinuous at $x = 2$ .

Check Condition 2 ( $\lim_{x \to 2} f(x)$ exists):

Factorise the numerator for $x \neq 2$ :

\lim_{x \to 2} \dfrac{(x - 2)(x + 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 2 + 2 = 4

The limit exists and equals $4$ .

Classification:

Because $\lim_{x \to 2} f(x) = 4$ exists but $f(2)$ is undefined, $f(x)$ has a removable discontinuity at $x = 2$ .

Example 2: Finding a Constant for Piecewise Continuity

Question: Find the value of the constant $k$ that makes the piecewise function $g(x)$ continuous for all real numbers:

g(x) = \begin{cases} kx^2 + 2x, & x \le 2 \\ 3x - 2, & x > 2 \end{cases}

Step-by-Step Solution:

Identify the boundary point: The transition occurs at $x = 2$ . Both sub-functions ( $kx^2 + 2x$ and $3x - 2$ ) are polynomials and thus continuous on their own intervals.

Calculate the left-hand limit and function value at $x = 2$ :

\lim_{x \to 2^-} g(x) = g(2) = k(2)^2 + 2(2) = 4k + 4

Calculate the right-hand limit at $x = 2$ :

\lim_{x \to 2^+} g(x) = 3(2) - 2 = 6 - 2 = 4

Equate limits for continuity:

4k + 4 = 4 \implies 4k = 0 \implies \mathbf{k = 0}

Example 3: Trigonometric Piecewise Continuity

Question: Find the value of $A$ such that the function $f(x)$ is continuous at $x = 0$ :

f(x) = \begin{cases} \dfrac{\sin(3x)}{x}, & x \neq 0 \\ A, & x = 0 \end{cases}

Step-by-Step Solution:

Evaluate the limit as $x \to 0$ :

\lim_{x \to 0} \dfrac{\sin(3x)}{x} = \lim_{x \to 0} \left( 3 \cdot \dfrac{\sin(3x)}{3x} \right)

Using the standard limit identity $\lim_{u \to 0} \dfrac{\sin u}{u} = 1$ :

\lim_{x \to 0} f(x) = 3(1) = 3

Set $f(0) = \lim_{x \to 0} f(x)$ :

A = 3 \implies \mathbf{A = 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

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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

What is a Continuous Function?
Algebra and Properties of Continuous Functions
Types of Discontinuity
Continuity in Piecewise Functions
Are Radical Functions Continuous?
Worked Examples & Solutions

Continuous Functions

What is a Continuous Function?

The Formal 3-Step Test for Continuity at a Point

Algebra and Properties of Continuous Functions

Standard Functions Continuous on Their Domains

Types of Discontinuity

1. Removable Discontinuity (Point Discontinuity)

2. Jump Discontinuity

3. Infinite Discontinuity (Essential Discontinuity)

Continuity in Piecewise Functions

Are Radical Functions Continuous?

Worked Examples & Solutions

Example 1: Testing Continuity at a Point

Step-by-Step Solution:

Example 2: Finding a Constant for Piecewise Continuity

Step-by-Step Solution:

Example 3: Trigonometric Piecewise Continuity

Step-by-Step Solution:

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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