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I got started recently on proofs about continuity and so on. So to start working with this on $n$-spaces I've selected to prove that every linear function $f: \mathbb{R}^n \to \mathbb{R}^m$ is continuous at every $a \in \mathbb{R}^n$. Since I'm just getting started with this kind of proof I just want to know if my proof is okay or if there's any inconsistency. My proof is as follows:

Since $f$ is linear, we know that there's some $k\in \mathbb{R}$ such that $|f(x)|\leq k|x|$ for every $x\in \mathbb{R}^n$, in that case let $a\in \mathbb{R}^n$ and let $\varepsilon >0$. Consider $\delta = \varepsilon /k$ and suppose $|x-a|<\delta$, in that case we have:

$$|f(x)-f(a)|=|f(x-a)|\leq k |x-a|<k \frac{\varepsilon}{k}=\varepsilon$$

And since $|x-a|<\delta$ implies $|f(x)-f(a)|<\varepsilon$ we have that $f$ is continuous at $a \in \mathbb{R}^n$. Since $a$ was arbitrary, $f$ is continous in $\mathbb{R}^n$. Is this proof fine? Or there was something I've missed on the way?

Gold
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  • The proof looks OK. The crucial step is showing the existence of $k$. Is this a theorem that you've already proved? – Ayman Hourieh May 24 '13 at 23:35
  • Sure, I've proved that result before, it's one of the exercises of Spivak's Calculus on Manifolds, and since it seemed pretty important I proved it before anything else. – Gold May 24 '13 at 23:37
  • Then you're done. The proof looks good. – Ayman Hourieh May 24 '13 at 23:39
  • Thanks for the feedback @AymanHourieh! – Gold May 24 '13 at 23:39
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    How did you proof the existence of $k$ without using the fact that linear transformations are continuous? – JKEG Mar 11 '18 at 21:36
  • @JKEG While applying Extreme Value Theorem is circular, Spivak proves it without using compactness by adding the lengths of the column vectors and multiplying by length. – RHyp Mar 29 '23 at 04:21

1 Answers1

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This proof is correct modulo result you stated in the begining, i.e. $$ \text{there exist $k\in\mathbb{R}$ such that $|f(x)|\leq k|x|$ for all $x\in\mathbb{R}^n$} $$ Proof of this fact is much more interesting and uses compactness of unit ball in $\mathbb{R}^n$.

Norbert
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