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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.

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Note that the first function is the distribution function of the uniform distribution over $[a,b]$. – Shai Covo May 19 '11 at 1:41
@Shai Covo It can be considered distribution function, but in other context. – Andrey May 19 '11 at 1:48
You can call it pretty much whatever you want... The function-naming committee has not yet come to a final conclusion. – Mariano Suárez-Alvarez May 19 '11 at 3:51
up vote 5 down vote accepted

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

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More precisely, a linear ramp. – lhf May 19 '11 at 1:20
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 1:23
please check my edit – Andrey May 19 '11 at 1:25
I have seen both being called that. Certainly yours would qualify. – Ross Millikan May 19 '11 at 2:31

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.

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

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You may call it that way since it is the integral of a Boxcar function. – Shai Covo May 19 '11 at 3:12

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