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A Flat Function This is a mathematical paper that is to be written as if it were to be put in a mathematical magazine. The instructions are on the attached page. The writer is to follow the instructions and do it as it says. This paper is to be written in word format and LaTex format. It is required that the paper is to be done in both of these formats. For this paper you will explore some properties of the function flat(x) = ( e − 1 x2 for x 6= 0 0 for x = 0 The audience for this paper is to be advanced undergraduate mathematics majors like yourself. You should use a tone suitable for a mathematics magazine for mathematics majors. You should use a format like: I. Introduction. II. Discussion. III. Properties. A. Lemma. Proof. B. Theorem. Proof. . . . VI. . . . Summary. References. 1. . . . 2. . . . . . . flat is a C∞(R) function, meaning that flat: R → R and possesses derivatives of all orders. The function flat is an example of a function whose Maclaurin series (Taylor series at 0) does not converge to it. flat and all its derivatives have value 0 at x = 0 and so its Maclaurin series is the identically 0 series, which corresponds to the identically 0 function. flat is clearly not identically 0. For instance, flat(1) = 1 e . Give proofs of the properties mentioned above, and discuss one or more additional topics related to the function. Some suggestions are given below, or you may find some on your own. 1 Suggested topic 1. By taking integrals of products of shifts of pieces of flat and the 0 function, normalizing and then taking products of shifted reflections, it is possible to produce a C∞(R) function with range [0, 1] and value one on an interval [−a, a] and value zero outside (−b, b) for 0 < a < b. This is called a C∞ bump function (at 0). A C∞ bump function at 0. From Tu [1], p. 130. Demonstrate how to construct a bump function like that above from flat. Suggested topic 2. You may know of the mathematical structures known as rings. One of the simplest rings is the ring of integers, Z. The C∞(R) functions also form a ring. These rings are, in fact, commutative rings since multiplication in them is commutative. Rings contain special subsets called ideals. For any subset S of a ring R, the intersection of all ideals of R containing S is an ideal I of R. This ideal is the smallest ideal of R containing S. It is denoted hSi, and is called the ideal generated by S. See Wikipedia [3] for further details. A desirable property of a ring is being Noetherian, which may be defined as having the property that any ideal is finitely generated, meaning that the ideal is hSi for some finite subset S of the ring. Being Noetherian means that the ring shares certain properties of the ring of integers, and is therefore “nice” in some sense. See Wikipedia [2]. The ring of C∞(R) functions turns out not to be Noetherian. This is because the ideal generated by the derivatives of flat is not finitely generated. A proof of this would be great, but a discussion of its plausibility would suffice.

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In basic calculus the primary functions we learn about are polynomials, the exponential function, logarithms, and trigonometric functions. Then we learn about Maclaurin series and more generally Taylor series and we are led to believe that for any function f(x) and any point a in its domain you can find an interval around a on which the function is equal to its Taylor series computed at a.

For many of the elementary functions this is indeed the case. There are a number of fine details to be checked: the function needs to be defined and infinitely differentiable for all Taylor series coefficients to be defined.
One also needs to determine the interval of convergence of the Taylor series. For most examples that are given this interval of convergence is not equal to zero.

Let us call a function "smooth" if the derivatives of all orders are defined at every point in its domain. We will study some smooth functions that have points where the
Taylor series at these points differ from the function itself:
the Taylor series predicts nothing about the function values.

We will start with the particular example of the function flat(x)

flat(x) = 0 for x <= 0
             = exp(-1/x) for x>0...
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