# Does the following function have a name and what properties does it have?

(Apologies for the vague way the question is stated.)

We define the function like this:

(1) $f(x) = x$ for $0\leq x \leq a$ (for some $a \geq 15 \in \mathbb{R}$) $\land$ $f(x) = a$ if $x > a$ . For $x<0$, it doesn't really matter what $f(x)$ looks like.

Has someone studied this type of function? Does it have a 'nicer' definition than the one stated above? Any references on the subject? It is not absolutely necessary for the function to have all of the above characteristics. If you know a function that is similar to it, please let me know.

Motivation: I'm looking for a function that assigns a constant to 'infinite values' of x and does not 'deform' finite values (too much). So the function should satisfy $f(\zeta(1))=a$ for some finite value of $a \in \mathbb{R}$ and $f(3)=3$, for example. It doesn't really matter whether $f(b)=b$ or any multiple of b or that $f(x)$ is some finite polynomial, as long as it's 'easy' to find the value of $f(b)$.

Thanks,

Max Muller

-
so f(x) = min(x, a)? – Qiaochu Yuan Oct 1 '10 at 16:10
What you're asking for in (1) looks like a flipped ramp function. I don't think this is what you really want, though. – J. M. Oct 1 '10 at 16:15
In electronics one can define cg.tuwien.ac.at/~theussl/DA/node18.html the "ramp signal" ramp$\left( x,w\right)$ as follows: ramp$\left( x,w\right) =1\quad$if $x\geq w$; ramp$\left( x,w\right) =x/w\quad$if $0\leq x<w$; ramp$\left( x,w\right) =0\quad$if $x<0$. Your function can be expressed in terms of the ramp function as $a$ ramp$\left( x,15\right)$ – Américo Tavares Oct 1 '10 at 16:22
@ all of the above, I guess you're all right. Judging from the information wolframalpha gave me and the links you provided me with , ramp(x,a) = min(x,a) in the positive half of the plane – Max Muller Oct 1 '10 at 16:55

f(x)=min(x,a)

We can rewrite this in terms of the absolute value function as (x+a)/2-|x-a|/2, the average minus half the difference. If you prefer this can be seen as getting the graph of the absolute value function, and manipulating it to fit the function you want. This is possible since they are both formed of two straight lines meeting at a point.

-