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Let K be a compact interval, J an arbitrary interval of R, and let f (x, y) be a continuous function on K × J. 2) ϕ (y) = D2 f (x, y)µ(x)dx. K Continuity and diﬀerentiability at a point y being local properties we can replace J in what follows by a compact interval H ⊂ J containing all points of J suﬃciently close to y. |µ(x)|dx ≤ M (µ) f K where M (µ) = |µ(x)|dx. Then ϕ(y) = µ(fy ) where fy (x) = f (x, y). 3) (|x − x | < r ) & (|y − y | < r ) =⇒ |f (x , y ) − f (x , y )| < r. e. 4) |y − y | < r =⇒ fy − fy K ≤ r.

To reduce to a Fourier series of period 1, one has to replace x by 2πx in the series cos x − cos 3x/3 + cos 5x/5 − . e. 8) = cos 2πx − cos(6πx)/3 + cos(10πx)/5 − . . = [e1 (x) + e−1 (x)] /2 − [e3 (x)/3 + e−3 (x)/3] /2 + . . 9) = −π/4 for 1/4 < |x| < 3/4, and by periodicity for the other values of x. 8), 1/4 ap = −1/4 = = = e−2πipx dx − 3/4 e−2πipx dx = 1/4 e−3πip/2 − e−πip/2 e−πip/2 − eπip/2 − = −2πip −2πip eπip/2 − e−πip/2 /2πip − e−πip eπip/2 − e−πip/2 /2πip = [1 − (−1)p ] sin(pπ/2)/πp, zero if p is even, and equal to 2(−1)(p−1)/2 /πp if p is odd; since we omitted a factor π/4, we ﬁnally have ap = 0 (p even) or (−1)(p−1)/2 /2p (p odd), which agrees with (8).

E. which is the upper envelope of a family of continuous real functions (which clearly excludes the value −∞); for example the function 1/x2 (x − 1)2 on R, with value +∞ for x = 1 or 0. For every a ∈ X and every M < ϕ(a), there is, by the deﬁnition of an upper bound, a continuous function f on X satisfying f (x) ≤ ϕ(x) for every x, f (a) > M. 7) ϕ(a) > M =⇒ ϕ(x) > M for every x ∈ X near a. This is the property which deﬁnes the lsc functions; equivalently, one may demand that, for every ﬁnite M , the set {ϕ > M } of the x ∈ X where ϕ(x) > M must be open in X since then, if it contains a, it must also contain all the points of X suﬃciently near a.