Function that returns 1 given any positive number

Question

What would be a way to write a function that given any positive number the function would return one, but for a negative number or zero it would return zero.

Also what mental process should I go through to approach such a problem?

For example:

f(45) = 1;

f(10) = 1;

f(1) = 1;

f(0) = 0;

f(-3) = 0;

Do you mean something different from "if ($x > 0$) then return 1; else return 0"? — Mitch, Dec 09 '16 at 23:45
You can use $F(x)=\frac {x+|x|}{2x}$ but you still need to handle $x=0$ as a special case. — lulu, Dec 09 '16 at 23:48
Perhaps it can be expressed as the pointwise limit of a sequence of continuous functions — MathematicsStudent1122, Dec 09 '16 at 23:48
$f_n(x) = e^{n(x - \vert x \vert)}$ converges pointwise to your function — Ethan Alwaise, Dec 09 '16 at 23:56
Are you saying that the solution should not be a piecewise function / should not define a special case? That is implied, but it isn't clear. — JeremyDouglass, Dec 09 '16 at 23:58

score 3 · Answer 1 · answered Dec 09 '16 at 23:50

3

It would be piecewise function like this $$f(n) = \begin{cases} 1, & \text{if $n > 0$} \\ 0, & \text{if $n \le 0$ } \end{cases}$$

answered Dec 09 '16 at 23:50

M. Rojas

27

score 2 · Accepted Answer · answered Dec 09 '16 at 23:51

2

$$f(x)=\lfloor\frac{\text{sgn}(x)+1}{2}\rfloor$$

answered Dec 09 '16 at 23:51

msm

7,147

Very nice! I found that one step further covers the zero case $$f(x)=\text{sgn}(\lfloor\frac{\text{sgn}(x)+1}{2}\rfloor)$$ – Simon The Cat Dec 10 '16 at 00:03
Hmm, actually I'm wrong my idea didn't work for zero since it returns 1 in that case. – Simon The Cat Dec 10 '16 at 00:17
Here is the definition of sgn$(x)$. – msm Dec 10 '16 at 00:49

Function that returns 1 given any positive number

2 Answers2