Principles of Mathematics in Operations Research
10.5 Web material 155 http://www-db.stanford.edu/~sergey/near.html http://www-history.mcs.st-andrews.ac.uk/Extras/Kuratowski_ To ...
11 Continuity In this chapter, we will define the fundamental notions of limits and continuity of functions and study the proper ...
158 11 Continuity Ve > 0,36 > 0 9 Vx 6 E with dx(x,p) < S we have du(f(x), f(p)) < s. Remark 11.1.3 The following ch ...
11.2 Continuity and Compactness k 1 £(TE). 1/2 = 6. Proposition 11.1.9 / is continuous if and only if every component is con- ti ...
160 11 Continuity Proof. Let g = f'^1 : Y -> X. Show that V closed set C in X, g~l{C) is a closed set in Y: 5 _1(C) = (/~^1 ) ...
11.4 Continuity and Connectedness 161 Example 11.3.4 f(x) = -,E = (0,1) C R. Let us show that f is not uni- formly continuous on ...
162 11 Continuity Proof. Assume that f{E) is not connected, i.e. 3nonemptyA,Bcy9Aflfl = f), AnB = 0, f{E) = A U B. LetG = EDf~^1 ...
11.4 Continuity and Connectedness 163 Fig. 11.2. Example 11.4.3 (i) If f(x+) or f(x—) does not exist, we say the discontinuity a ...
164 11 Continuity then for this specific e > 0, 35 > 0 9 W with x < t < x + 5, we have \f(t)-f(x)\<e. CASE 1: X G ...
11.5 Monotonic Functions 165 f(x-) f(x) f(x+) f ^0 Fig. 11.4. Proof of Theorem 11.5.3 Corollary 11.5.4 Monotonic functions have ...
166 11 Continuity f(t) > Cm + /(«)• Fix f(s) => cm + f(s) is a bound for all f(t)'s. So, take the infimum over t's. f(xm+) ...
11.6 Web material 167 Fig. 11.5. A heated wire index.html http://at.yorku.ca/course/atlas2/node7. html http://at.yorku.ca/i/a/a/ ...
168 11 Continuity Theorem.ppt http://whyslopes.com/Calculus-Introduction/Theorem- One_Sided_Range.html http://www-history.mcs.st ...
12 Differentiation Differentiation In physical terms, differentiation expresses the rate at which a quantity, y, changes with re ...
170 12 Differentiation Remark 12.1.4 The converse is not true. One can construct continuous functions which fail to be different ...
12.2 Mean Value Theorems 171 Theorem 12.2.3 Suppose f : [a,b] H->- R is differentiable and f'(a) < A < f'(b) [/'(a) > ...
172 12 Differentiation 12.3 Higher Order Derivatives Definition 12.3.1 // / has a derivative /' on an interval and if f is itsel ...
Web material where Rr-\(x,h) is the remainder. Furthermore, mr 1 Problems 12.1. Suppose / : [0, oo) H-» K is continuous, /(0) = ...
174 12 Differentiation http://en.wikipedia.org/wiki/Mean_value_theorem http://en.wikipedia.org/wiki/Taylor's_theorem http://grus ...
13 13 Power Series and Special Functions In mathematics, power series are devices that make it possible to employ much of the an ...
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