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I have a group of questions where I basically need to show that $f(x)$ is continuous from $\Bbb{R}^n$ to $\Bbb{R}$.

Honestly, I'm not sure how to approach this generally.

The first one seems to obvious: $f(x_1,x_2,...,x_j,...,x_n)=x_j$ but I have no idea on how to write a proof for this.

The next one: $f(\mathbf{x})=|\mathbf{x}|$. I know that $|x|$ is continuous from $\Bbb{R}$ to $\Bbb{R}$, but how do I prove this here?

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    Use the same proofs of $\varepsilon$-$\delta$ for functions from $\mathbb R$ to $\mathbb R$, use chessmath's hint to deduce what $\delta$ works for $\varepsilon$.2012-04-22
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    The answer depends on what you understand by "continuity". And possibly, on what you understand by "convergence".2012-04-22

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