Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i\_1$ upto $i\_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i\_n$ and $n$ is odd, and $0$ if $n$ is even.

What is $\lim\_{t \rightarrow \infty} f(t,1/3)$?  

What is $\lim\_{t \rightarrow \infty} f(t,1/2)$?  

What is $\lim\_{t \rightarrow \infty} f(t,1/\pi)$?  

What is $\lim\_{t \rightarrow \infty} f(t,x)$?

joriki clarification in comments is correct, does $\lim\_{t \rightarrow \infty} f(t,1/\pi)$ exist, is it 0 or 1 or (0 or 1) or undefined?
Is it incorrect to say that is (0 or 1)?

Is there a way to express this:  

$K=\lim\_{t \rightarrow \infty} f(t,x)$

K, without limit operator ?

I think to say K is simply undefined is an easy way out. Something undefined cant have properties. Does K have any properties? Is K a concept?