First, Hailu's explanation is correct in any single word. It says what the restriction of a function, by definition, is.

I would like to say something to the question: why to consider restrictions of functions?

One of the basic ideas in modern mathematics is that functions are individuals, single objects like pi is a single number. So to be sure: when we restrict a function, we get a new function (in a convenient manner, since we have not to repeat the main stuff, namly how f(x) is defined for given x). Consider for instance f : R--> R, x |--> 1/x . f is obviously not continuous (as a whole, which means on its domain, which here is R). If you experiment with some constuction that needs a continuous function as input, you can't take f since it isn't coninuous. You could try the restriction of f to R\{0} (which most people denote f | R\{0} ). This function is continous and even differentiable as often as you want. You can use this function in deductions and theorems either under the name f | R\{0} or under a simple name, say h, upon a definition h := f | R\{0} again as a new function. Of course, in my example the formula for the function value was very short (namely 1/x). You may have experienced situations in mathematics, or (even more so) in physics and engineering, where function defining formulas are much longer and more complicated to write, and here the restriction notation pays off.

This touched only the most obvious aspect of the question, ' why to consider function restrictions?'. Hope it helps.

Thanks a lot.I was expecting something like this,as Mr.Ulrich Mutze explained.

The definition of the concept restrictions of a function is very simple:

Let f : X→Y be a function that transform an element x of the set X into a new element y of a set Y; this means y = f(x), where xϵX and yϵY. It could happen that the function f could not be defined for all elements xϵX. In this case, you need to define a subset A of X in which the element f(x) is always defined. Then the restriction of the function f to A is defined as another function defined in the following form:

If A is a subset of X, then the restriction of the function f to A is the function f|A: A→Y; this means f|A(x) = y, where xϵA and yϵY

In physics, we have to restrict our functions quite often to conserve its physical sense. For example, mass (or temperature) has to be positive (or non-negative at least), while the formula containing them may be well defined for negative mass too. The restricted function is indeed different from the original one: it is simply udefined whenever mass is negative.

@Jorge.

> and > doesn't go together! It is clear what you wanted to express, but whoever takes your words as they stand can't but be confused. Besides this confusion, I can't see that you said something that wasn't said before.

My intención with my reply was not to say something new because there is not to much to say to explain this simple definition. What I did was to use some words to explain the definition of restricción of a function plus some mathematic expressions.

In my explanation I did not indicated that the función f is defined in the whole set X and for this reason I introduced the subset A which is in my example the real domain of the función f.

An example f: X→Y y=f(x)=1/x donde X=R and Y=R

In this case R is not the domain of the function f but R\{0} and Y is not the image of f but R\{0}.

A function g from X' to Y is a restriction of the function from X to Y iff X' is included in X and for all x of X', g(x) = f(x).

That's the definition. A function need not be writable as an expression, it is only a possible shorthand. A function "is" (is equivalent to, is isomorphic to ...) a set of ordered pairs, whose first element belongs to a set X and the second one to the set Y, which is not necessarily different from X. A restriction is simply a subset of that set.

@Ulrich, 1/x is not defined at 0 too, therefore continuity has no meaning at that point.

@Claude: Sorry for having been sloppy. For me it is second nature that any inversion is understood as Penrose's pseudo-inversion and this means 1/0 is defined and has the value 0. I forgot to mention this when I replaced my first intended example sqrt(|x|) and missing differentiability by 1/x and missing continuity.

@Jorge: My point was that when you write f: X --> Y you say that f(x) is defined for every x in X. Otherwise the notation f: X --> Y would be useless.

Well, In my expalnation I didn't say that f(x) is defined for every x in X. What I did say is xϵX, not for all x. The notation f: X→Y does not mean that f is defined for all x, just that the function f takes an element x and transform it in an element y.

@Jorge: Give me a reference to a single (reasonably modern and reasonably serious) math text book in which f : X -->Y does not imply dom(f) = X. Then I will give you a dozen in which it does.

Dear Ulrich

The question here is not if I have hundreds of books that support my point of view on this issue or if you have thousands of books that support your position. The question here as mathematicians is how you define the expression f: X→Y. If you say that f is a function that transform an element xϵX into an element f(x)ϵY, then you cannot say that the domain of the function f is X. This can be true but also cannot be true. However, if you say that the function f transform all elements xϵX into an element f(x)ϵY, then you can say without doubt that the domain of the function f is X.

In my explanation I did not say that the function f transform all element xϵX into an element f(x)ϵY. For this reason, I introduce the concept of “restriction of the function f (f|A)”.

It is a question of the “definition of the expression” used something common in Mathematics.

Dear Jorge,

>

I insist in telling you that the community of textbook writers and Professors in Mathematics since at least 50 years agree on this definition (in the sense I indicated). It is true that in physics and engineering a lot of writing is done without any care for such formal things. So you find in published articles a lot of examples for your floppy usage.

You might be interested to hear that in Germany there are DIN standards (national form of ISO standards) for the proper usage of set theoretical expressions. DIN 5473 p. 8 says that the meaning of f : A --> B is:

f is a function (defined earlier, of course)

and domain(f) = A

and f(A) contained in B.

German engineers are expected to follow this definition.

Having gone in my life through some thousends of pages of mathematical proofs, I have some experience how many hours you can loose with digging through proofs that do not use a reasonably well defined notation. And so I feel a reponsibility for not confusing the young part of our auditorium by ignoring well established notational standards.

@Ulrich: German engineers are in comfortable situation then, lucky guys. Is the same true for pure mathematicians? My feeling is that Jorge's point of view is not completely separated from reality ... Will you indeed protest seeing something like this:

f: R --> R, with f(x)=1/x

?

@Marek: unfortunately I wrote just this myself in my first contribution to this thread. Claude noticed this inconsistency and I explained this with my tacid understanding that 1/0 = 0 which is a wide-spread point of view among computation-oriented guys. Without such an understanding I would indeed protest loudly. Such floppyness ruines clear communication! I admitted that one can find such blunder in printed articles on a regular basis ... in reality.

Dear Ulrich

Perhaps mathematicians in Germany and maybe in other countries follow a strict definition of the expression f: X→Y in the sense you explain but this not means that in all places and all mathematicians should follows that definition. I am such mathematician. What I like of Mathematics is that you can, respecting some general rules and principles, define how you are going to use some mathematical expressions. What you cannot do is to define a mathematical expression violating this rules and principles.

Dear Jorge,

to avoid such inconclusive arguments was the intention when I asked for a math text book as witness for your relaxed understanding of f: X --> Y. When you now install yourself as such a witness: ' I am such mathematician' , this proves nothing.

Dear Ulrich

I have no intention to prove nothing when I say I am a mathematician. I just want to explain my answer that is all. You have the right to agree or disagree with my reply. I have the same right. This is not an academic tribunal in which you are defending a doctoral thesis or the result of a research, this is just an exchange of opinion about a specific question submitted by another colleague and everybody should have the right to present his or her point of view. I am not going to continue with this issue.

When the domain is equal to the starting set, that is also called a "mapping" or "map" ("Abbildung" auf Deutsch, "application" en français.) So that standard is at least useless.

The restriction has not the only purpose of getting a function with some nice property, it is a mathematical relation on its own. For example, it is a partial order. Another example that has important applications in geometry and topology, is the restriction of a function from a surface (or higher dimensional space) to one from a curve. That's the way the partial differentiation is defined.

If f: A -> B is a function from A to B, and C is a subset of A, the restriction of f to C is the function g from C to B such that g(x) = f(x) for all x in C. This function can be represented by the symbol (f|C).

The restriction of a function to a subset of its domain is a function with that subset as its domain and having the same effect as the original function .

If f is a function that maps set X to a set Y and Z is a subset of X, then the restriction of f to Z is the function f |Z : Z -> Y such that Z(x) = y and y is an element in Set Y. Sometimes we need to define a new domain for X.

@Steve,

does your Z(x) result from a typing error or is it it meant as such? Then, what is its intended meaning?

Steve

In your answer, do u mean " f|Z(x)=f(x)  for all x in Z "  instead of  "Z(x)=y" ?

I just notice that we always discussed restriction of f: X->Y as a modification of X. There is, however, a second kind of restriction which is concerned with a modification of Y. If Y' is a set such that f(X) \subseteq Y' \subseteq Y, then f defines in a natural way a mapping X -> Y' which is said to be the restriction of f to the codomain Y'. Notice that there is always such a set, namely Y' = f(X). Restriction to this particular codomain always results in a surjective mapping. The codomain-restriction is not discussed in too many textbooks.