By Daniel A. Murray

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Extra resources for A First Course in Infinitesimal Calculus

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By a strip we mean a set of the form W(y; a, b), where a, b are finite. If h E 1i (]RN) and h( rx) -+ 0 as r -+ +00 uniformly for xES, then h == 0 by the maximum principle. The following example shows that uniformity cannot be dispensed with. 7. 3) lim IlxWh(x) = 0 X --;

5. 7). Then (1. 9) lim sup If (x):::; lim sup J(z) (y E oo D). x-+y,xED z-+y,zEaD a Further, if J is continuous in the extended sense at y E aD, then If (x) --* J(y) as x --* y; also If(x) --* limo-+oo J(z) as x --* 00 if J has a limit (finite or infinite) at 00 . Proof. 9) is established the rest of the theorem will follow by applying this inequality to J and - J. 9), we suppose that limsuPz-+yJ(z) < A < +00 and show that lim sUPx-+y If(x) :::; A. We treat the cases y E aD and y = 00 separately.

This can be seen in the case where y = 0 by observing that a set E in ]RN is a sphere or hyperplane if and only if E = {x E ]RN : allxl1 2 + (x, z) + {3 = OJ, where a and {3 are real numbers, z E ]RN and IlzW > 4a{3 . Under inversion with respect to 5(0, a) the image of this set E is E* = {x E ]RN\{O}: {3l1xll 2 + (x,a 2 z) +a4 a = OJ. This establishes the claim with y can apply a translation . 3. If f E C2(n) and f* is the image of f under the Kelvin transform with respect to 5, then <1f*(x) = IIxll- 2- N (<1J)(x*) (x E nO).

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