A function whose value is either one or zero

Question

First I apologize in advance for I don't know math's English at all and I haven't done math in almost a decade.

I'm looking for a function whose "domain/ensemble of definition" would be ℝ (or maybe ℤ) and whose "ensemble/domain of variation" would be ℕ{0, 1} that would look like something this awesome ascii graph...

          ^f(x)
          |          
          |
1________ | __________
0________\./__________>x
         0|
          |
          |
          |

f(x) is always 1, but 0 when x = 0

Actually I need this for programming, I could always find other ways to do what I wanted with booleans so far but it would be much better in most cases to have a simple algebra formula to represent my needs instead.

I just hope it's not too complicated a function to write/understand and that it exists of course. Thanks in advance guys.

Answer 1

The function $f: \mathbb{R} \rightarrow \{0,1\}$ defined via $$ f(x) = \lim_{n \rightarrow \infty} \sqrt[n]{x^2} $$ satisfies $f(0) = 0$ and $f(x) = 1$ for all $x \neq 0$. This is entirely useless for programming, but it is the simplest function I can think of that avoids a piecewise definition.

Answer 2

Define $f: \mathbb{R} \rightarrow \{0,1\}$ via $$ f(x) = \begin{cases} 0 &\text{ if } x = 0\\ 1 &\text{ if } x \neq 0. \end{cases} $$

If you are programming, this can be accomplished with a single if-then-else statement. This will surely be more efficient to compute than any "algebra formula" (by which I take you to mean "non-piecewise formula"), since if clauses are built into the language at a very low level.

Answer 3

I think that the most compact way to write this is using the Iverson Bracket:

$f: \mathbb{R} \to \{{0,1}\}$

$$ f(x) = [x \neq 0]$$

Answer 4

I found this question relevant in social science when you are testing for the effect of a variable but are uncertain whether it is its mere presence which is significant, or whether the degree of the presence should be used. I have used the following approach in Excel to convert degree of presence ($A_2 \ge 1$) to just presence ($A_2=1$).

The following simple nested 'IF' statement will return 1 if $A_2$ is a positive number, $0$ for $0$, and $-1$ for a negative number.

=IF(A2=0,0,IF(A2>0,1,-1))