Thanks for the response.  I want to add to the question that the function need not be differentiable at x=0. And yes, algebraic means polynomial or rational. The function may contain mod(x) as well.

Thank you very much sir. This expression may work. But may I ask again that if a single polynomial in |x| will be able to do the same?

An algebraic function is a function that can be defined as the root of a polynomial equation. Functions that can be constructed using only a finite number of elementary operations (involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power) together with the inverses of functions capable of being so constructed are examples of algebraic functions. Nonalgebraic functions are called transcendental functions.

Perhaps the simplest function that fulfills the conditions imposed is the absolute value of x, since:

==> It is algebraic.

==> It is symmetrical about the Y axis (even function).

==> It vanishes at the origin.

==> Its slope at the origin is +/- 1.

Thank you Peter for your corrections.

Clearly, the definition given in Wikipedia (English version) is nonsense as it confuses a function with a number. The truth is that I've simply copied and pasted it hastily,without reading it. I have also not read what was added after the statements in bold: "The function Reaches asymptotically to slope = 0 as x goes away form zero.".

I made a chess movement without thinking.... I apologize for that.

Peter, your arguments are correct, however I think it was not necessary to write so much about something that was clearly a lapsus.

Of course, my intention was not a boast of erudition but simply help who asked for help (although the attempt proved to be certainly unfortunate).

Let's see if I'm now more successful.

Classifying the set of all functions in two categories, algebraic and transcendental, functions, algebraic, y = f (x), I think that could be defined as those whose matching rule is an algebraic expression, that is, an explicit algebraic function is one whose dependent variable, and is obtained by combining a finite number of times the independent variable x with real constants by algebraic operations of addition, subtraction, multiplication, division, raising to power and root extraction.

So far, we are proposing functions involve the absolute value of x, │x│. Actually, y = │f (x) │ is a compact way to define a piecewise, ie, y = f (x) if f (x) = 0.

Therefore, we are admitting piecewise functions and in this case, it is possible that a polynomial function fulfills all the conditions imposed.

Por example is y = f (x) the function defined as follows:

f (x) = 1 for -inf

Sorry... where it says "the pending 0 should be equal on both sides, which contradicts the condition that they are +1 on one side and -1 on the other". meant "the slope at 0 should be equal on both sides, which contradicts the condition that they are +1 on one side and -1 on the other."

O.K., Peter. Thank you for all.

Best regards.

Thanks for the responses.

Best regards from Spain Raushan.

Well, for a while now I'm living in Malaga (southern Spain), but I'm from Asturias (northern Spain). Málaga is called "Costa del Sol" but Asturias is "The Narural Paradise ;).

Are you working in Madrid?.

Once again, sorry; Rereading my reply, I observe typographical errors: are missing some minus signs. The piecewise function that I tryed to suggest as an example should be this:

Let y = f (x) be the function defined as follows:

f (x) = 1 for -inf < x

I'm spanish; in English: This is a polynomial function that satisfies all imposed conditions.

Peter, your definition of my function as f(x) = min(|x|, 1) has the beauty of simplicity. I like it!.

Yes, is a long way from Asturias to Málaga.

I suggest you see some pictures of Asturias, for example:

http://www.laspain.com/cose-da-vedere-immagini-monumenti/asturias/somiedo/388.html#.VyvAkISLTIU

http://www.apartamentosruralesllanes.es/blog/las-playas-del-oriente-de-asturias-llanes-y-ribadesella/