Mathematical Analysis I by Frédéric Cao

By Frédéric Cao

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Example text

The logarithm is strictly increasing if a > 1, strictly decreasing if a < 1 (Fig. 19). The previous properties translate into the following: 1 iogcti^y) -= log^ ^* -f logo y. yx,y>Q, X iogr l o g ^ ^ - - loga y. 19. , the circle of unit radius centred at the origin O = (0,0), whose 52 2 Functions equation reads X'^ + F^ = 1. Starting from the point A = (1,0), intersection of the circle with the positive x-axis, we go around the circle. More precisely, given any real x we denote by P{x) the point on the circle reached by turning counter-clockwise along an arc of length x if x > 0, or clockwise by an arc of length —X if X < 0.

2. 2) (bottom right) Among piecewise functions, the following are particularly important: v) the a b s o l u t e v a l u e (Fig. 3, top left) f{x) /: X if X > 0, —X if X < 0; vi) the s i g n (Fig. 3, top right) / : R - -Z, r+1 f{x) = sign(a;) = 1 0 l-l ifa;>0, ifa; = 0, ifa;<0; vii) the i n t e g e r p a r t (Fig. 3, b o t t o m left), also known as floor f u n c t i o n . ] = - ! 3. Clockwise from top left: graphs of the functions: absolute value, sign, mantissa and integer part viii) the mantissa (Fig.

Let / be defined on X with values F . If / is one-to-one and onto, it is called a bijection (or bijective function) from X to Y. If so, the inverse map j ~ ^ is defined on y , and is one-to-one and onto (on X); thus, f~^ is a bijection from Y toX. For example, the functions /(x) = ax + 6 (a ^ 0) and /(x) = x^ are bijections from R to itself. , from [0,+oo) to [0,+(X))). If / is a bijection between X and F , the sets X and Y are in bijective corrispondence through / : each element of X is assigned to one and only one element of F , and vice versa.

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