# Download A First Course in Sobolev Spaces by Giovanni Leoni PDF

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By Giovanni Leoni

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Let e > 0. 5 (u (E)) + e. 1. Monotone Functions 16 Since u is continuous, the set A := u 1(V) fl (a, b) is open. Define the function v (x) := u (x) - Rx, x r= [a, b]. Then D_v (x) > 0 for every x E E, and hence E C Am, where A,, is the set defined in the previous lemma. Let {(ck, dk)} be the family of all connected components of A. 3) R (dk - CA,) < u (dk) - u (ck) . ) . 3), and (L1) once, in this order, we obtain R1. (A,,,) = R (dk - Ck) < (u (dk) - u (CO) k k < L o (u (Au)) = Lo U (u (ck) , u (dk)) , k where we have used the fact that intervals (u (ck) , u (dk)) are pairwise disjoint, since u is increasing.

Uns (x) - Un, (x0) X - xo x - xo :=1 It follows that lim sup zyzo z=i x - anf - (xo - an,) = 1. x - xo u (x) - u (xo) > x - XO for every 1 E N. Hence, lim sup zPzo u (x) - u (xo) = oa, x - xo which implies that u is not differentiable at xo. 36. 17)) is not differentiable contains the set E, but in general it may be larger. 2. 41 below for some properties of this set). We refer to [27] and [171] for more information on this subject. 37. Let I C R be an interval and let u : I - lR be a monotone function.

E. x E [a, b]. Step 2: If I is an arbitrary interval, construct an increasing sequence of intervals [an, such that b,/'supl. Since the union of sets of Lebesgue outer measure zero still has Lebesgue outer measure zero and since on each interval [an, bn] the set of points in which u is not differentiable has Lebesgue outer measure zero by the previous step, it follows that the set of points of I in which u is not differentiable has Lebesgue outer measure zero. 30. The extended real numbers 2-u (x), 17-u (x), D+u (x), and D+u (x) are called Dini's derivatives of u at x.