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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 $0$ ?

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2 Answers 2

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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@Jebrunho Thanks for your comment. Though I did not understand your answer clearly, but a related question of mine is what is the definition of the domain of function? Does domain include the points where function takes $\infty$ value? –  Abdul kadir Dec 5 '12 at 3:52
The domain is essentially whatever you choose. For instance, consider a machine $f$, which receives whatever value I give it and spits out another. The set of all values that I give it is the domain. Sometimes, for a function to be well-defined, we must restrict the domain. However, this does not seem to be the case for your function. And the domain may or may not include the points where the function is $\infty$, depending on how you define the domain. –  Jebruho Dec 5 '12 at 3:56
@Jebrunho I know these. But my question was, is it 'mathematically' correct if we include \infty in the domain? I am asking about maths, not about any specific application. Maths should not have 'may or may not answer' (I know I can discard bad points from my application :) ) –  Abdul kadir Dec 5 '12 at 4:22
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. –  Jebruho Dec 5 '12 at 4:27

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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