Point-to-Set Maps and Mathematical Programming by P. Huard

By P. Huard

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32. (a) f(x) = sgn (x). (b) f(x) = [x]. 19 Example (Parabolas). , a # 0, are functions whose graphs (in an orthonormal Cartesian frame) are parabolas. , since x 2 ~ 0 \:Ix E JR.. Actually the range of f(x) = x 2 , x E JR, is [0, +00[. This last claim deserves a few more words. First it states that for every y < 0 there is no x E JR such that x 2 = y, which is trivial. But it also states that for each y ~ 0 there is an x E JR such that x 2 = y; a solution of the last equation is the square root vY of y.

The function absolute value or norm defined by f(x) = lxi, x E JR, has [0, +oo[ as range and is not injective, d. 31. 21 Example. The circle with center at (0,0) and radius r > 0, is the union of the graphs of the two functions f+(x) = ~, x E [-r,r], and f-(x) = -~, x E [-r, r], with ranges respectively [0, r] and [-r, 0]. f + and f _ are not injective. 22 Example. Similarly, the ellipse with semiaxis a, b > 0 centered at (0,0) is the union of the graphs of the two functions f + (x) = bJ1 - x 2 / a 2 , x E [-a, a], and f _ (x) = -bJI - x 2 /a 2 , x E [-a, a].

Be the . ", "let A := {x E lR I x 2 < 2}" which reads "Consider the set A of real numbers with square less than 2", or There exists a ... such that . as in "Given a straight line l' and a point P not in 1', there is a point l' such that the line through P and Q is perpendicular to r" . Q in These declarations usually hold inside the context for which they have been made. For instance, if we declare a constant in a proposition, we can use it in its proof. c. Variables There is also the need to use labels for objects belonging to a specific class, as in Let x be a real number.

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