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Unformatted text preview: 8.1 A familx of nonanalytic functions. Let m 2 0 be any nonnegative integer. Define el/X2
——7H—— for x # O,
x
fm(x) =
0 for x = 0. We will show that each of the functions fm(x) has continuous
derivatives ofall orders, for all x. We also show that none
of them is analytic near 0; that is, none of them equals a
power series of the form 2 anxn‘ in an interval about ’0. Theorem 1. (a) The function fm(x) is continuous for (b) Furthermore, f$(x) exists for all x and satis fies the equation f$(x) = —mfm+l(x) + 2fm+3(x). Proof. (a) The general theorem about composites of continuous functions shows that fm(x) is continuous when x # 0. To prove continuity at x = 0, we must show that el/x2>
lim —————— = 0. m
x+0 x The substitution u = l/x2 simplifies the calculation. We have 2
]_/x 1.l III/2
' §____. = ' e = ' E...
lim m 11m ——ﬁ7§ 11m u . x+0 x u+m l/u u+w e This limit is zero because eu approaches infinity faster than any power of u, as p __> w. (b) We check differentiability. If x # 0, we calculate directly: 2 as = m +3. ewi—zJ
X To show the derivative exists at x = 0, we apply the defini— tion of the derivative: f (0+h)  f (0)
f$(o) = lim _E_____H__JE___
h+0
—1/h2 m 1/h2
= lim (e /h )  O — li E—————
h _ m m+l
h+0 h+0 h This limit is zero, by part (a). Therefore, the derivative exists at x = 0 and equals 0. Thus the formula f$(x) = ~mfm+l(x) + 2fm+3(x) holds when x = 0.
Theorem 2. The function fm(x) has continuous deriva tives of all orders, for all x, but fm(x) does not egual . n .
a power series 2 anx 9g anx interval about 0. Proof. _We know that each function vfm(x) is differen tiable, for all x. The equation £51m) = —mfm+l(x) + 2fm+3(x) shows us that f$(x) is differentiable, for each x. This is the same as saying that derivative f$(x) exists for all x.
In general, we proceed by induction. Suppose we are given that the nEll derivative of each function fm(x) exists, for all x. Then the preceding equation shows that the nEE derivative of the function f$(x) also exists, for all x. This is the same as saying that the (n+l)§E derivative of
fm(x) exiSts. It follows that the nt—ll derivative of fm(x) exists, for all x and all n. And of course it is continuous because the (n+l)§E derivative exists. Now we suppose fm(x) = E anxn on some nontrivial interval about x = 0, and derive a contradiction. If fm(x) equals this power series, then the coefficients an must satis. fy the equations (n)
fm (0)
a = ———— n n! for all' n. We know that fm(x) vanishes when x = 0. Using the equation
f$(x) = —mfm+l(x) + 2fm+3(x) repeatedly, we see that all the derivatives of fm(x) also vanish at x = 0. Therefore an = 0 for all n, so fm(x) is_identically zero in some interval about x 0. But this is not true; indeed the function’ fm(x) vanishes onlz for x = O. ...
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 Spring '09
 BRUBAKER
 Derivative, #, 9g, 11m, Nell, 2

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