I am afraid that this question goes against the norm of this site. But, I came across an exercise in my textbook which intrigued me. The question goes as follows:
If $\phi (x) = \lim _{n \to \infty} \frac{f(x)+\cos^{2n}{\pi x}\{2g(x)-f(x)\}}{1+\cos^{2n}{\pi x}} $, prove that $\phi(x) = g(x)$ or $\phi (x) = f(x)$ according as $x$ is an integer or not.
The proof is pretty straightforward using the fact that $ 0 \leq\cos^{2}{ \pi x}< 1$ when $x$ is not an integer, and $=1$ when it is an integer.
But my question is, can you use the above function (with suitable modification) as an indicator function under some circumstances (e.g. Here we can view this function as an indicator whether or not $x$ is an integer) ?!
P.S: I am aware of the fact that this question has no direct answer and it is indeed a bit novice question.
From the comments above.
Sure, you can consider it as an indicator function. But whether it is ever useful to take this to be an indicator function is. . .doubtful. I agree that the limit itself is interesting, but I just don't see any applications of this function under the limit. If I only need the indicator function of the integers, then taking $1_{\mathbb{Z}}$ serves my purpose just as well as the limit of this function.