Continuous function

A continuous function maps nearby inputs to nearby outputs. This article explains the intuitive idea, the epsilon–delta definition, common variants, examples, and why continuity matters in analysis and topology.

Author: Leandro Alegsa Created: November 13, 2022 Updated: June 27, 2026

A continuous function is a basic concept in mathematical analysis that formalizes the intuitive idea that small changes to the input produce only small changes to the output. Informally, the graph of a continuous real function looks like an unbroken curve that can be drawn without lifting the pen. More precisely, continuity can be stated in several equivalent ways depending on the setting: pointwise conditions, uniform conditions, and topological formulations. The general notion applies to functions between metric or topological spaces as well as the familiar case of functions on the real line. $x_{0}$ A mathematical function that fails this property at some point is called discontinuous there.

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Formal (epsilon–delta) definition

The classical epsilon–delta formulation, attributed to early 19th-century analysts, describes continuity at a point x0 in the domain of a function f. Given any tolerance ε > 0 on the change in output, there must exist a corresponding tolerance δ > 0 on the change in input such that whenever the input x satisfies |x − x0| < δ, the outputs satisfy |f(x) − f(x0)| < ε. This definition is usually presented for functions on the real numbers but extends naturally to metric spaces. $f(x_{0})$ A standard historical reference credits Augustin-Louis Cauchy for bringing rigor to these ideas. $x_{0}$ The epsilon–delta condition formalizes the phrase "arbitrarily small" that appears in informal descriptions. $\varepsilon >0$

Basic properties and common facts

Operations such as addition, subtraction, multiplication and composition preserve continuity: sums and products of continuous functions are continuous on common domains. $\delta >0$
Continuous functions on closed and bounded intervals have strong consequences: they attain maxima and minima and satisfy the intermediate value property (they take every intermediate value between two outputs). $|x-x_{0}|<\delta$
Continuity can be considered at a point, on an interval, or globally on a domain; uniform continuity is a stronger, domain-wide version of the epsilon–delta condition.

Variants: stronger or weaker notions

Several refinements of continuity are useful in analysis. Lipschitz continuity imposes a linear bound on how fast a function can change: there exists a constant L such that |f(x) − f(y)| ≤ L |x − y| for all x,y in the domain. This condition guarantees uniform continuity and often simplifies estimates in differential equations and numerical analysis. $|f(x)-f(x_{0})|<\varepsilon$ Hölder continuity relaxes Lipschitz to allow power-law bounds. Uniform continuity requires a single δ for each ε that works everywhere on the domain, rather than depending on the point. $x_{0}$ In topology, continuity is equivalent to the preimage of every open set being open, which generalizes the notion far beyond the real line. $x_{0}$

Graphical intuition is often helpful: a continuous real-valued function has no sudden jumps, removable holes, or infinite spikes at the point in question. Simple discontinuities include removable discontinuities (a hole in the graph), jump discontinuities (different one-sided limits), and essential or infinite discontinuities (unbounded behavior). A common visual test is the "no-lift" rule: if you can trace the graph from left to right without lifting your pen, the function is continuous on that interval. $L$

Continuity is central across mathematics: it underpins differential calculus, ensures the solvability of equations via the intermediate value theorem, and is a fundamental concept in topology and functional analysis. Historical figures such as Cauchy and Weierstrass helped make the definition precise and explored subtleties such as functions that are continuous everywhere but differentiable nowhere. Elementary tools like the absolute value function appear repeatedly in continuity arguments and proofs. $|f(x)-f(y)|\leq L|x-y|$ Understanding continuity and its variants is essential for studying limits, derivatives, integrals, and the behavior of mathematical models in science and engineering.

Motivation

The function

$f(x)={\begin{cases}x,&{\text{wenn }}x\leq 1\\x+1,&{\text{wenn }}x>1\end{cases}}$

"jumps" at the point x=1 from the function value 1 to the function value 2. If the function represents a connection from nature or technology, such behavior appears unexpected (Natura non facit saltus). If, for example, the function describes the relationship between the energy expended in cycling and the speed achieved, it would be surprising if a minimal increase in the energy expended at one point led to a sudden doubling of the speed.

The mathematical notion of continuity attempts to accurately describe those functions that do not have such "arbitrary" behavior. Thus, the given function is not continuous, where the discontinuity can be restricted to the point . x=1 In all other points the function is continuous.

Continuity is often associated with being able to draw the graph of a function in one go, without having to drop it. This view reaches certain limits, especially when considering functions with other ranges of definition than the entire real number line. Therefore mathematically exact definitions are needed.

For example, the

$g(x)={\begin{cases}x\cdot \sin {\frac {1}{x}},&{\text{wenn }}x\not =0\\0,&{\text{wenn }}x=0\end{cases}}$

given function is clearly continuous, because except at x=0 its graph is a continuous line, and at x=0 it has no place to make "jumps". But whether it can be "drawn without settling" to the zero point cannot be decided without a more precise definition of what an allowed drawing should be. There it is easier to develop a definition of "steady" without the notion of "drawing", according to which this function can be proved to be steady. Then the reasons just mentioned may well contribute to the proof.

Detailed study of the behavior of g(x) near the point x=0 : Drawing possible?

The function is even, so it suffices to restrict the investigation to its behavior for $x\geqq 0$ Besides the zero at the zero point, its positive zeros are at the points $x_{k}={\tfrac {1}{k\pi }}$ for integer k>0 ; these are indexed from right to left, i.e. $x_{1}>x_{2}>\dotsb >0$ with $x_{1}$ as the largest zero and with each infinitely many other zeros between $0$ and any other zero $x_{k}$ . Between the adjacent zeros $x_{k}$ and $x_{k-1}$ lies one place ξ $\xi _{k}={\tfrac {2}{(2k-1)\cdot \pi }}$ with , $\sin {\tfrac {1}{\xi _{k}}}=\pm 1$ such that $|g(\xi _{k})|=\xi _{k}={\tfrac {2}{(2k-1)\cdot \pi }}>{\tfrac {1}{k\pi }}$ . Thus, between the zeros $x_{k}$ and the graph must $x_{k-1}$ twice ${\tfrac {1}{k\pi }}$ overcome the height difference its length in this section is more than ${\tfrac {2}{k\pi }}$ Between the zero $x_{1}$ and any other positive zero $x_{n}$ further left, the length of the graph is thus greater than ${\tfrac {2}{\pi }}\cdot ({\tfrac {1}{1}}+{\tfrac {1}{2}}+\dotsb +{\tfrac {1}{n-1}})$ . This sum grows as increases across all bounds (see Harmonic Series), so the complete drawing would never be finished.

Sine and cosine are continuous functions, their function graphs can be drawn in one go without stopping.

Continuity of real functions

Definition

Let be a real function, that is, a function $f\colon D_{f}\to \mathbb {R}$ , whose function values are real numbers and whose domain of definition $D_{f}\subset \mathbb {R}$ also consists of real numbers.
In real analysis, there are several equivalent ways to define the continuity of in an ${\displaystyle x_{0}\in D_{f}}$ . The most common are the epsilon delta criterion and the definition using limits.

Definition by epsilon delta criterion. is called continuous in $x_{0}$ if for every ε $\delta >0$ there exists $\varepsilon >0$ a δ such that for all $x\in D_{f}$ with

$|x - x_0| < \delta$

applies:

$|f(x) - f(x_0)| < \varepsilon$ .

Intuitively, the continuity condition means that for every change ε $\varepsilon$ in the function value that one is willing to accept, one can find a maximum change δ $\delta$ in the argument that ensures this default.

Example: Prove continuity of the function $f(x)=2x+3$ at the point $x_{0}$

Let $x,x_{0}\in \mathbb {R}$ and ε $\varepsilon \in \mathbb {R}$ with

$\varepsilon >0$ .

It's

$|f(x)-f(x_{0})|=|(2x+3)-(2x_{0}+3)|=|2x-2x_{0}|=2|x-x_{0}|$ .

To make this smaller than the given number ε $\varepsilon$ , for example.

$\delta (\varepsilon )={\tfrac {1}{2}}\varepsilon$

be chosen. Because from

$|x-x_{0}|<\delta ={\tfrac {1}{2}}\varepsilon$

then follows namely

$|f(x)-f(x_{0})|=2|x-x_{0}|<2\cdot {\tfrac {1}{2}}\varepsilon =\varepsilon$ .

Comments:

Since the function at every point $x_{0}\in \mathbb {R}$ is continuous, is thus continuous on all of $\mathbb {R}$ continuous.
Because δ $\delta$ can only be derived from ε $\varepsilon$ , but not from the location $x_{0}\in \mathbb {R}$ is even continuous on all of $\mathbb {R}$ uniformly continuous.

Definition by means of limit values. With this definition one demands the interchangeability of function execution and limit value formation. Here one can either rely on the notion of limits for functions or for sequences.
In the first case one formulates: is called continuous in $x_{0}$ if the limit $\lim _{x\to x_{0}}f(x)$ exists and $f(x_{0})$ coincides with the function value i.e. if holds:

$\lim _{x\to x_{0}}f(x)=f(x_{0})$ .

In the second case, one formulates: is said to be continuous in $x_{0}$ for every sequence $x_{0}$ convergent to $(a_{n})$ with elements $a_{n}\in D_{f}$ , the sequence ${\bigl (}f(a_{n}){\bigr )}$ $f(x_{0})$ converges to
The second condition is also called the sequence criterion.

Instead of continuity in $x_{0}$ one often speaks of continuity in the point $x_{0}$ or continuity in the place $x_{0}$ . If this condition is not satisfied, then called discontinuous in (at the point/place) $x_{0}$ , or denote $x_{0}$ as the discontinuity point of .

A function is said to be continuous if it is continuous at every point in its domain.

Examples of continuous and discontinuous functions

If a function is differentiable at one point, it is also continuous there. Thus follows in particular the continuity

of all rational functions (so, for example, $x\mapsto x^{3}+4x^{2}-6x+9$ or $x\mapsto {\frac {2x-1}{x+2}}$ )
of the exponential functions $x \mapsto a^x$ , for fixed $a\in \mathbb {R} ^{>0}$
the trigonometric functions (i.e. sine, cosine, tangent,...)
of the logarithm functions

But the continuity of these functions can be proved directly without recourse to the notion of differentiability.

The absolute value function is also continuous, even if it is not differentiable at the point 0. Also continuous are all power functions (for example $x\mapsto x^{\frac {1}{2}}={\sqrt {x}}$ ), although they are also not differentiable for an exponent less than 1 at the point 0.
In fact, all elementary functions are continuous (for example, $x\mapsto {\sqrt {1+\cos ^{2}(x-5)}}$ ).

When considering elementary functions, however, it should be noted that some elementary functions have as their domain of definition only a real subset of the real numbers. For example, the square root function omits all negative numbers, and the tangent function omits all zeros of the cosine.
In these cases, it is sometimes imprecisely formulated that the functions are discontinuous in the corresponding places. However, this is not correct, since the question of continuity only arises for points in the domain of definition. Mathematically meaningful, however, is the question of a continuous continuation of the function at a definition gap.
For example, the function

$x\mapsto {\frac {2x-1}{x+2}}$

defined for all real numbers $x\not =-2$ and continuous in every point of its domain. It is therefore a continuous function. The question of continuity in does x=-2 not arise because this point is not part of the domain of definition. A continuous continuation of the function at this definition gap is not possible.

The absolute value function and the root function are examples of continuous functions that are not differentiable at individual points in the domain of definition. At the beginning of the 19th century, mathematical experts still assumed that a continuous function must be differentiable at least at "most" places. Bernard Bolzano was then the first mathematician to construct a function that is continuous everywhere but differentiable nowhere, the Bolzano function. However, he did not publish his result. In the 1860s, Karl Weierstrass also found such a function, known as the Weierstrass function, which caused a sensation in the mathematical world. The graph of the Weierstrass function can effectively not be drawn. This shows that the intuitive explanation that a continuous function is one whose graph can be drawn without setting down the pencil can be misleading. Ultimately, one must always resort to the exact definition when investigating the properties of continuous functions.

Using methods of 20th century mathematics, it could even be shown that the functions which are nowhere differentiable are in a sense "frequent" among the continuous functions.

Simple examples of discontinuous functions are:

the sign function (discontinuous only in 0)
the Dirichlet function (discontinuous at every point)
the Thomaean function (discontinuous exact in all rational numbers).

Continuity of composite functions

Similar to differentiability, continuity is a property that transfers from the components to the functions composed of them in many operations. For the following points, let the continuity of in $x_{0}$ already be given.

Consecutive execution: if is another real function whose domain of definition spans the range of values of and is $f(x_{0})$ continuous in then the composition is $g\circ f$ continuous in $x_{0}$ .
Algebraic operations: If is another real function with the same domain of definition as , which is also continuous in $x_{0}$ , then the pointwise defined functions , $f-g$ , $f \cdot g$ and $\tfrac{f}{g}$ also continuous in $x_{0}$ . In the latter case, however, note that the domain of definition of the composite function turns out to be $D_{f}$ without the zero set of . In particular, $x_{0}$ may thus not be a zero of even in this case.
Maximum/minimum: Under the same conditions as in the previous point, the pointwise defined functions $\max(f,g)$ and are $\min(f,g)$ continuous in $x_{0}$ .

If the definition ranges of the functions involved do not match as required, you may be able to help yourself by placing suitable restrictions on the definition ranges.

Under certain conditions continuity also transfers to the inverse function. However, the statement cannot be formulated here for point-wise continuity:

If the domain of the injective continuous real function an interval, then the function is strictly monotone (increasing or decreasing). The inverse function defined on the range of values of $f^{-1}$ is also continuous.

Using these permanence properties, one can derive, for example, the continuity of the elementary function given above $x\mapsto {\sqrt {1+\cos ^{2}(x-5)}}$ from the continuity of the cosine, the identical function, and the constant functions. Generalizing this reasoning, the continuity of all elementary functions follows as a consequence of the simple examples given earlier.

Main theorems about continuous real functions

There are a number of important theorems that hold for continuous real-valued functions These are most easily formulated by assuming that $D_{f}=[a,b]$ with $a,b\in \mathbb {R} ,a<b$ is a closed bounded interval:

Intermediate value set: the function takes any value between and .

Theorem of Minimum and Maximum: is bounded and infimum and supremum of its function values are also assumed to be function values. So it is really a matter of minimum and maximum. This theorem proved by Weierstrass, sometimes called the extreme value theorem, only provides the existence of these extreme values. For their practical finding, statements from the differential calculus are often necessary.

Fundamental theorem of analysis: is Riemann integrable and the integral function

$F(x)=\int _{a}^{x}f(t)\,{\rm {d}}t$

is a primitive function of .

Heine's theorem: satisfies a stricter version of the epsilon-delta criterion. The corresponding property is called uniform continuity.

From the intermediate value theorem and the theorem of minimum and maximum together it follows that the image of is also a closed, bounded interval (or, in the case of a constant function, a single-point set).

Other continuity terms

Tightening of the concept of continuity are, for example, uniform continuity, (local) Lipschitz continuity, Hölder continuity as well as absolute continuity and geometric continuity. Ordinary continuity is sometimes called pointwise continuity to distinguish it from uniform continuity. Applications of Lipschitz continuity can be found, for example, in existence and uniqueness theorems (e.g., Picard-Lindelöf's theorem) for initial value problems of ordinary differential equations and in geometric measure theory. Absolute continuity is used in stochastics and measure theory, geometric continuity in geometric modeling.

One property that a set of functions may possess is uniform continuity. It plays a role in the frequently used theorem of Arzelà-Ascoli.

Example for the sequence criterion: The sequence exp(1/n) converges to exp(0)

Author

AlegsaOnline.com Continuous function Leandro Alegsa

URL: https://en.alegsaonline.com/art/22777

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@misc{alegsaonline_22777,
  author = {Alegsa, Leandro},
  title = {Continuous function},
  year = {2026},
  howpublished = {AlegsaOnline.com},
  url = {https://en.alegsaonline.com/art/22777},
  note = {Updated: June 27, 2026; Language: en}
}

TXT

Leandro Alegsa. “Continuous function.” AlegsaOnline.com. Updated: June 27, 2026. https://en.alegsaonline.com/art/22777