Continuity in R^n

Builds on Euclidean space Limit of a function of several variables

Definition

Let $D \subseteq \mathbb{R}^n$ , let $f \colon D \to \mathbb{R}^m$ , and let $a \in D$ . The function $f$ is continuous at $a$ when, for each $\varepsilon > 0$ , there is a $\delta > 0$ such that $\|f(x) - f(a)\| < \varepsilon$ for each $x \in D$ with $\|x - a\| < \delta$ . Here $\|\cdot\|$ is the Euclidean norm of Euclidean space. The function $f$ is continuous on $D$ when it is continuous at each point of $D$ .

Remarks

Let $a$ be a limit point of $D$ . Then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$ , with the limit of a function of several variables. A function into $\mathbb{R}^m$ is continuous if and only if each component function is continuous. Sums, products and compositions of continuous functions are continuous. So each polynomial in $x_1, \dots, x_n$ is continuous on $\mathbb{R}^n$ . Continuity in each variable separately is weaker: see the separately continuous function that is not continuous.

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Euclidean space object
Function concept
Function of several variables concept
Limit of a function of several variables concept
Continuity in R^n definition

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134 pages build on this one.

Tag	`0S9EK5MQPE`
URI	`m2://0S9EK5MQPE`
Kind	definition
Writing	W0 stub
MSC	26B05
Defines	Continuity in R^n
Uses	Euclidean space, Limit of a function of several variables

Continuity in R^n

Definition

Remarks

Reading path

Where this leads

Further reading

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