By Edgar Asplund; Lutz Bungart
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Additional resources for A first course in integration
X; and H W S Œ0; 1 ! s// for every s 2 S . If f is a covering map, then there exists a continuous function HO W S Œ0; 1 ! s/ for every s 2 S . If S is connected, then the mapping HO is unique as a consequence of the following theorem. 5 (, Chap. 2, Theorem 2). Let X Â IRn , Y Â IRm , and Z Â O HO W Z ! X be two continuous IRk be sets, f W X ! Y be a covering map, and let G; O O O mappings with f ı G D f ı H . z/ for some point O O z 2 Z, then G D H . To formulate the main theorem, we need some additional notions.
Q /. F /. If the intersection of f . / and f . F /, then both polyhedra f . / and f . U / of every sufficiently small open neighborhood U of x0 . This, however, contradicts the openness of f and thus f . / and f . Q / are contained in different halfspaces which shows f . / and f . F /. Thus f is coherently oriented. 1 that coherent orientation does not imply injectivity. The following slight modification of this example shows that surjectivity does not imply openness. 2. 0; 1/; and let f be the piecewise linear function which coincides on the set conefvi ; vi C1 g with the linear map that carries vi onto wi and vi C1 onto wi C1 , i D 1; 2; 3; and which coincides with the identity outside of the union of these cones.
They are concerned only with the behavior of a function in a neighborhood of a given point. For this purpose it is convenient to have a local approximation concept at hand. 2 Basic Notions and Properties 27 Suppose f :IRn ! IRm is a piecewise affine function with corresponding polyhedral subdivision ˙ and let x0 2 IRn be some fixed point. x0 /j. In particular, we can find for every vector v 2 IRn a real number ˛0 > 0 such that ˛0 v 2 U . x0 / is well defined and positively homogeneous. 22) which is continuous.