What is this function called?

Question

What is this function called? Here is a graph:

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$f(x, a, b) = 0$, if $x < a; \quad \quad \displaystyle \frac {(x - a)} {(b - a)}$ if $a < x < b; \quad \quad$ and $1$ if $ x > b$.

Closest name is Sigmoid function, but they all are smooth. Also I remember I saw similar formula for saturation, but not sure about that, googling didn't help.

Edit: Ramp function is closer:

enter image description here

But it is general one, while my is more specific one, used for normalization.

Note that the first function is the distribution function of the uniform distribution over $[a,b]$. — Shai Covo, May 19 '11 at 01:41
@Shai Covo It can be considered distribution function, but in other context. — Andrey, May 19 '11 at 01:48
You can call it pretty much whatever you want... The function-naming committee has not yet come to a final conclusion. — Mariano Suárez-Álvarez, May 19 '11 at 03:51

score 6 · Accepted Answer · answered May 19 '11 at 01:18

6

I call it a ramp function. At least some others do.

answered May 19 '11 at 01:18

Ross Millikan

374,822

2

More precisely, a linear ramp. – lhf May 19 '11 at 01:20
Ramp function (at least what I found) is first two parts of my function. Important point here is that my function range is strictly in [0; 1] and used for normalization, where ramp is general one. May be there is more specific name? – Andrey May 19 '11 at 01:23
please check my edit – Andrey May 19 '11 at 01:25
I have seen both being called that. Certainly yours would qualify. – Ross Millikan May 19 '11 at 02:31

score 4 · Answer 2 · edited Nov 21 '12 at 00:06

The first two "pieces" of your graph seem to fit the "ramp" function, with $(a,0)$ replacing $(0,0)$ of the "general" example you provide for a ramp function. But the first and third "pieces" comprise what could be described as a step-function (with endpoints joined by the second "piece"/line).

But in the end, since it seems we need to define and discuss three separate "pieces" of the function, depending on the value of x with respect to the parameters $a$ and $b$, perhaps the most accurate (albeit general) description for your function is simply a piecewise-linear function, (consisting of a linear-ramp function which continues/plateaus at its maximum value, $y=1$ for all $x>b$?). I can't seem to find any more concise description that captures all you'd like it to capture.

Shai Covo · Answer 3 · 2011-05-19T02:37:50.223

2

Such a function falls in the category of Elementary special functions. You may call it a (normalized) integrated Boxcar function.

edited May 19 '11 at 02:37

answered May 19 '11 at 01:40

Shai Covo

24,077

You may call it that way since it is the integral of a Boxcar function. – Shai Covo May 19 '11 at 03:12

What is this function called?

3 Answers3