Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $0$ ?

share|improve this question
add comment

2 Answers 2

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.

share|improve this answer
add comment

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 $\infty$.

share|improve this answer
    
@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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.