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If there is a subset $S$ in $\mathbb{R}^n$ consider the characteristic function $\chi_S: \mathbb{R}^n \to \mathbb{R}$.

i) what would be the value of $\chi_S(p)$ for any $p \in \mathbb{R}^n$? Attempt at understanding: I understand that the characteristic function means that it would be 1 at the points in $S$ and 0 at the points not in $S$, but I guess I'm not really sure what this question is asking me. If it's a random point that's in $\mathbb{R}^n$ but not in $S$ then the value would be 0, but if it's in $\mathbb{R}^n$ and $S$ (since $S$ is a subset of $\mathbb{R}$) it would be 1. So how can you tell if it's 1 or 0? Do I write both? I feel like the question is only asking me for one value.

ii) Also prove that this characteristic function $\chi_S$ is discontinuous at ALL the boundary points in $S$. I'm not sure how to go about this, but after thinking about it I think I am supposed to negate the epsilon delta definition of continuity and use it to prove it. I'm not quite understanding WHY this is true so I'm having trouble starting it even though I know I have to use it.

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First step: forget about the generality of $n$-dimensional space, and look at the case $n=1$. Even let $S=[0,1]$, unit closed interval. Now draw the graph, identify the boundary of $S$, and see why the function is discontinuous at those two points. Recast your proof so that it makes sense in more general situations. Next try for $S$ the set of rational points in the real line, do the same. Proceed from simple examples to complicated ones, and finally get a proof valid for all cases. – Lubin Nov 1 '12 at 2:40

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