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Indicator function is defined for a set $C$ as,

$\delta_{C}(x) =\begin{cases} 0 & \text{if } x \in C \\ \infty & \text{if } x \not\in C \end{cases}$

Now what is the domain of this indicator function? Is it the same set where the function value is $0$ ?

2 Answers 2

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The question depends upon whether you are considering $\infty$ to be a number, and over what set you are defining your function. Usually, we do not consider $\infty$ a number, and thus the domain would exclude all places where $\delta(x)=\infty$, that is, $C$. However if you are considering $\infty$ to be a number, your answer would include all places from which you are willing to draw $x$, so probably the reals, but possibly any set.

With that in mind, I'm going to go ahead and say that the answer is probably $C$, because $\infty$ is probably not a number.

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If it is the same, then the $\infty$ would not be needed. It seems like you would choose the domain. For instance, let the domain be $\mathbb{R}$ and then let $C=\mathbb{Q}$. Then the function on the rational numbers would be 0 and on the irrational numbers would be $\infty$.

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    Maths does have may or may not answers. Regardless, if your domain is the extended reals, then the extend reals contain $\infty$ so that the domain would contain $\infty$. But, once again, it depends on the domain you choose.2012-12-05