Continuity¶

Definition¶

Let $f\colon X\to\mathbb{R}$ and $x_0\in X\text{.}$ We say that $f$ is continuous at $x_0$ if for every $\varepsilon>0$ there exists $\delta>0$ such that, for every $x\in X\text{,}$

|x-x_0|<\delta\implies|f(x)-f(x_0)|<\varepsilon.

The function is continuous on $X$ if it is continuous at every point of $X\text{.}$

Interpretation¶

Continuity expresses the idea that small changes in the input of a function lead to small changes in its output. At a point $x_0$ this means: however close to $f(x_0)$ we want the value $f(x)$ to be, we can achieve it by taking $x$ close enough to $x_0$ .

In the definition the tolerance $\varepsilon$ comes first and is arbitrary; then a $\delta$ must be found, and it may depend on $\varepsilon$ and on $x_0$ . Geometrically, the part of the graph above $(x_0-\delta,x_0+\delta)$ must lie in the horizontal band of height $2\varepsilon$ around $f(x_0)$ . Unlike the limit at a point, the point $x=x_0$ itself is included and the target is the actual value $f(x_0)$ , so $f$ must be defined at $x_0$ .

Here the band is centred at $f(x_0)$ itself and $x=x_0$ is allowed. Shrink $\varepsilon$ : for $x^2$ a $\delta$ can always answer, but for the moved point and the jump no $\delta$ works once $\varepsilon$ is small.

Continuity at $x_0$ can fail in several ways, matching the kinds of singularity: the limit exists but differs from $f(x_0)$ , the one-sided limits are different, as for the signum function at $0$ , or the values become unbounded or keep oscillating. The Dirichlet function is continuous at no point at all.

A common slip is to take "can be drawn without lifting the pen" as the definition. Continuity is checked point by point, and only at points of the domain: $1/x$ is continuous, as the Examples explain, although its graph consists of two separate pieces. The pen picture fits continuous functions on an interval, the setting of the intermediate value theorem and the extreme value theorem.

Math III later replaces $|x-x_0|$ by a general distance in continuity between metric spaces.

Key Results¶

If $x_0$ is a limit point of $X\text{,}$ continuity at $x_0$ is equivalent to the limit at that point satisfying $\displaystyle\lim_{x\to x_0}f(x)=f(x_0)\text{;}$ endpoints use the corresponding one-sided limit.
Sums, differences, products, quotients (where defined), maxima and minima of continuous functions are continuous.
If $\displaystyle\lim_{x\to x_0}g(x)=L$ and $f$ is continuous at $L\text{,}$ then $\displaystyle\lim_{x\to x_0}f(g(x))=f(L)\text{,}$ so a continuous function may be interchanged with a limit.
Standard facts used from here onwards. Each of the following is continuous at every point of its domain: polynomials, and rational functions wherever the denominator is nonzero; the power functions $x\mapsto x^{\alpha}$ on $(0,+\infty)$ and $\sqrt[n]{\cdot}$ on its natural domain; the exponential functions $x\mapsto b^{x}$ and the logarithms $x\mapsto\log_b x$ for $b>0\text{,}$ $b\neq1\text{;}$ the trigonometric functions $\sin$ and $\cos$ on $\mathbb{R}\text{,}$ and $\tan$ wherever $\cos x\neq0\text{;}$ and the absolute value $x\mapsto|x|\text{.}$

Examples¶

The function $f(x)=x^2$ is continuous on $\mathbb{R}\text{.}$
The function $f(x)=1/x$ is a continuous function: it is continuous at every point of its domain $\mathbb{R}\setminus\{0\}\text{.}$ The point $x=0$ lies outside the domain; there the values become unbounded, which is classified as an infinite discontinuity of $f$ (see Singularity), but this does not make $f$ a discontinuous function.

Graphs of x, x squared, x cubed, 1/x, the square root of x and e to the x; each is continuous on its domain, and the graph of 1/x breaks only at 0, which is not in the domain.

Prerequisites¶

Limit at a Point

Sources¶

Math I: Calculus · Lecture 5

Keywords: continuous function Symbols: alpha, delta, element of, epsilon, infinity, real numbers, square root

Last updated: 20 October 2026 03:02