Re: Differentiability at a point: understanding starting from set theory; request for help.
- From: "Mkajuma" <mkajumap@xxxxxxxxxxx>
- Date: Tue, 26 Apr 2005 03:22:18 GMT
Adam,
You seem as confused as I was (sometimes I think I am even more confused
than before). I used a tutor whose website is at www.mathtutor1.com . This
guy is a college professor and explain things well. He does tutoring live
online, but believe it or not the equipment he uses makes it better than
meeting someone in person. I don't know if he still offers the first 30
minutes for free but he did for me and I bet if you ask he will do the same
for you.
Good luck,
Mkajuma
"A. Boom" <aboom@xxxxxxxxxx> wrote in message
news:Z_Wdnac-0bKbPfDfRVn-2Q@xxxxxxxxxxxxx
> Dear readers,
>
> I seek to understand a few ideas: derivatives at a point, if the domain
> is a subset of a finite set; the implicit assumptions in the following
> definitions.
>
> I acknowledge that my understanding is lacking and hence I am asking
> readers for assistance. I've been learning applied math and seek a more
> thorough understanding of these calculus ideas from a rigorous
perspective.
>
> I began with top definitions and then proceeded backwards to undercover
> what they rely on and what is meant. It seemed logical to do so.
>
> Ultimately, my goal is to understand the sequence of ideas leading from
> the axioms of set theory to the definition of a derivative at a point.
>
>
>
> 1) Definition. A function is differentiable at a if f'(a) exists. It is
> differentiable on an open interval (a,b) [or (a,infinity) or
> (-infinity,a) or (-infinity, infinity)] if it is differentiable at every
> number in the interval.
>
>
>
> So the above definition relies at the definition of what "f'(a) exists"
> means.
>
>
>
> 2) Definition. The derivative of a function f at a number a, denoted by
> f'(a), is
> f'(a) = lim_{h -> 0} (f(a+h) - f(a))/h
> if this limit exists.
>
>
>
> The above definition requires the definition of a limit.
>
>
>
> 3) Definition. Let f be a function defined on some open interval that
> contains the number a, except possibly at a itself. Then we say that the
> limit of f(x) as x approaches a is L, and we write
> lim_{x -> a} f(x) = L
> if for every number epsilon > 0 there is a corresponding number delta >
> 0 such that
> |f(x) - L| < epsilon whenever 0 < |x - a| < sigma.
>
>
>
> At last the fundamental definition is reached, or so it seems. However,
> the above definition relies on what "|f(x) - L|" means. It is to be the
> absolute difference between two numbers. Yet, that seems to imply that
> "|f(x) - L|" be meaningful for elements of the range of the function,
> and "|x - a|" be meaningful for the elements of the domain of the
> function. Is this always true? (I assume not). Is it implicitly assumed
> that numbers from the real set are being used for the domain and
> codomain since they have such properties?
>
> The above definitions are from the book titled Calculus: Single
> Variable, Early Transcendentals by Stewart.
>
> Definition 3) relies on, at least, the definition of a function, as well
> as the meaning of the magnitude function(?).
>
> (Note: I'm aware that some authors treat set relations before functions,
> however that order need not be followed, as taken from Proofs and
> Fundamentals: A first course in abstract mathematics by Bloch.)
>
>
>
> 4) Definition. Let A and B be sets. A function (also known as a map) f
> from A to B, denoted f: A -> B, is a subset F <= AxB such that for each
> a in A, there is one and only one pair of the form (a,b) in F. The set A
> is called the domain of the function and the set B is called the
> codomain of the function.
>
>
>
> The definition given above does not state requirements of the sets used
> as the domain and codomain of a function. As such, why does it seem to
> be assumed that the sets have the property of being ordered in the
> definition for derivatives. That is, where x > y or x < y or x = y,
> which is required for the use of "|f(x) - L".
>
> When elements of a set are listed, the order of listing is subject to
> the writers whim, and not dictated by the set, as far I know. A set can
> be shown to follow a relation such as given above.
>
> Why is it that when derivatives are discussed, even represented as
> graphs, the elements of the domain are ordered, when the elements of the
> set may be arranged so, but do not have such ordering? Like 1, 2, 3, 4,
> 5, and not 3, 5, 2, 1, 4.
>
> I read that the pioneers of differential calculus had as their main
> problem that of "instantaneous change", which was reasonably thought of
> as a real world problem. As such, the domain, being time, was
> necessarily ordered.
>
> Is the concept of a derivative not applicable to unordered sets?
>
> The last questions are the most difficult for me to clearly state, and
> perhaps that means they are silly; I don't know.
>
> Let A be a set where A = {1, 2, 3, 4, 5}. Let B be a set where B = {2,
> 4, 6, 8, 10}. Now let f: A -> B be a function with elements (1,2),
> (2,4), (3,6), (4,8), and (5,10). This function, when plotted, has an
> ordering and a simple graph. Does it have a derivative at any or all
> points of A? If not, why?
>
> Define a second function by g: A -> B with elements of g being (1,4),
> (2, 2), (3, 10), (4, 8), (5, 6). A plot of the graph does not have a
> "smooth" shape. Does this function have a derivative at any or all
> points in its domain? If not, why?
>
> Are derivatives not defined for arbitrary sets? Must the elements of the
> sets, themselves, be elements of sets such as the real numbers so that
> there is always a number between any two elements, even if such elements
> aren't in the domain of the function?
>
> I suppose that the last three paragraphs highlight what I don't
understand.
>
> Any and all constructive help is appreciated.
>
> Thank you for your time, Adam.
.
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