Definition:Continuous Real Function

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This page is about continuous mappings in the context of real analysis. For other uses, see Definition:Continuous Mapping.

Definition

Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.

Let $x \in A$ be a point of $A$.


Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \to x} f \left({y}\right)$ exists and:

$\displaystyle \lim_{y \to x} \ f \left({y}\right) = f \left({x}\right)$


Continuous Everywhere

Let $f : \R \to \R$ be a real function.


Then $f$ is everywhere continuous if and only if $f$ is continuous at every point in $\R$.


Continuity on a Subset of Domain

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.


Then $f$ is continuous on $A$ if and only if $f$ is continuous at every point of $A$.


Continuity from One Side

Continuity from the Left at a Point

Let $x_0 \in A$.

Then $f$ is said to be left-continuous at $x_0$ if and only if the limit from the left of $f \left({x}\right)$ as $x \to x_0$ exists and:

$\displaystyle \lim_{\substack{x \mathop \to x_0^- \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.


Continuity from the Right at a Point

Let $x_0 \in S$.

Then $f$ is said to be right-continuous at $x_0$ if and only if the limit from the right of $f \left({x}\right)$ as $x \to x_0$ exists and:

$\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^+}$ is a limit from the right.


Continuity on an Interval

Where $A$ is a real interval, it is considered as a specific example of continuity on a subset of the domain.

Open Interval

This is a straightforward application of continuity on a set.

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.


Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$ if and only if it is continuous at every point of $\left({a \,.\,.\, b}\right)$.


Closed Interval

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Definition 1

The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is:

$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the right at $a$
$(3): \quad$ continuous on the left at $b$.

That is, if $f$ is to be continuous over the whole of a closed interval, it needs to be continuous at the end points.

Because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined.


Definition 2

The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is continuous at every point of $\left[{a \,.\,.\, b}\right]$.


Half Open Intervals

Similar definitions apply to half open intervals:

Let $f$ be a real function defined on a half open interval $\left({a \,.\,.\, b}\right]$.

Then $f$ is continuous on $\left({a \,.\,.\, b}\right]$ if and only if it is:

$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the left at $b$.


Let $f$ be a real function defined on a half open interval $\left[{a \,.\,.\, b}\right)$.

Then $f$ is continuous on $\left[{a \,.\,.\, b}\right)$ if and only if it is:

$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the right at $a$.


Real-Valued Vector Function

Let $\R^n$ be the cartesian $n$-space.

Let $f: \R^n \to \R$ be a real-valued function on $\R^n$.


Then $f$ is continuous on $\R^n$ iff:

$\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: d \left({x, a}\right) < \delta \implies \left|{f \left({x}\right) - f \left({a}\right)}\right| < \epsilon$

where $d \left({x, a}\right)$ is the distance function on $\R^n$:

$\displaystyle d: \R^n \to \R: d \left({x, y}\right) := \sqrt {\left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)}\right)}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right)$ are general elements of $\R^n$.


Informal Definition

The concept of continuity makes precise the intuitive notion that a function has no "jumps" or "holes" at a given point.

Loosely speaking, a real function is continuous at a point if the graph of the function does not have a "break" at the point.


Historical Note

The concept of a continuous real function was pioneered by the work of Carl Friedrich Gauss, Niels Henrik Abel‎ and Augustin Louis Cauchy.


Also see


Generalizations


Sources