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?

  • $\begingroup$ 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 $\varepsilon$. $\endgroup$ – Asaf Karagila Apr 22 '12 at 23:10
  • $\begingroup$ The answer depends on what you understand by "continuity". And possibly, on what you understand by "convergence". $\endgroup$ – André Caldas Apr 22 '12 at 23:25

Hint: These functions are Lipschitz, that is $|f(x)-f(y)|\leq |x-y|$.


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