# 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

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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. –  Guess who it is. 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