Why unit vector function is continuous?

Question

I wonder why $\dfrac{x}{\|x\|}$ is a continuous function, $x$ is a vector in $\mathbb{R}^{n}$. I have read in this site that the quotient of two continuous functions is continuous, but in books, there is no theorem that says that in case that we have a scalar valued vector function multiplied by a vector function.

Can anyone tell me the rule I'm missing?

Thanks in advance.

This is continuous in its domain, i.e., in $\Bbb R^n\setminus{0}$. — Hagen von Eitzen, Jun 19 '17 at 08:58
You can view it component wise and it becomes scalar valued function divided by scalar valued function. — mathreadler, Jun 19 '17 at 08:58
Each component is a quotient of continuous functions. Maybe not in books this particular result, but it's remarked often the idea of the components as functions $\mathbb R\to\mathbb R$ — Rafa Budría, Jun 19 '17 at 08:59

Emilio Novati · Accepted Answer · 2017-06-19T12:12:10.720

2

As a first step prove that a vector valued function is continuous if and only if all its components are continuous functions. ( You can see: Continuity of a vector function through continuity of its components). Then note that , for any component $x_i$, the function $\frac{x_i}{||x||}$, for $||x|| \ne 0$ is continuous as a quotient of continuous functions.

edited Jun 19 '17 at 12:12

answered Jun 19 '17 at 09:24

Emilio Novati

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Why unit vector function is continuous?

1 Answers1