By Judith L. Gersting
Skillfully conceived and written textual content, with many targeted good points, covers features and graphs, directly traces and conic sections, new coordinate structures, the spinoff, styles for integration, differential equations, even more. Many examples, routines and perform difficulties, with solutions. complex undergraduate/graduate-level. 1984 variation.
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Extra info for Technical Calculus with Analytic Geometry (Dover Books on Mathematics)
Example text
Any multimap F : X ⇒ Y induces a mapping from P(X) into P(Y ) (still denoted by F) given by F(A) := F(a), a∈A so that F({a}) = F(a). Multimaps can be composed: given F : X ⇒ Y, G : Y ⇒ Z, the composition of F and G is the multimap G ◦ F : X ⇒ Z given by (G ◦ F)(x) := G(F(x)), where G(B), for B := F(x), is defined as above. Then one has the associativity rule H ◦ (G ◦ F) = (H ◦ G) ◦ F. The inverse F −1 : Y ⇒ X of a multimap F : X ⇒ Y is the multimap given by F −1 (y) := {x ∈ X : y ∈ F(x)} , y ∈ Y.
Any multimap F : X ⇒ Y induces a mapping from P(X) into P(Y ) (still denoted by F) given by F(A) := F(a), a∈A so that F({a}) = F(a). Multimaps can be composed: given F : X ⇒ Y, G : Y ⇒ Z, the composition of F and G is the multimap G ◦ F : X ⇒ Z given by (G ◦ F)(x) := G(F(x)), where G(B), for B := F(x), is defined as above. Then one has the associativity rule H ◦ (G ◦ F) = (H ◦ G) ◦ F. The inverse F −1 : Y ⇒ X of a multimap F : X ⇒ Y is the multimap given by F −1 (y) := {x ∈ X : y ∈ F(x)} , y ∈ Y.
The first assertion being immediate, let us establish the second one. Let us set f (w) := lim infx→w f (x) and g(w) := lim infx→w g(x). If f (w) = −∞, or if g(w) = −∞, the result is obvious. Otherwise, given r < f (w), s < g(w), we can find U,V ∈ N (w) such that inf f (U) > r, inf g(V ) > s, whence inf( f + g)(U ∩ V ) > r + s. It follows that lim infx→w ( f + g)(x) ≥ r + s. Lower semicontinuity can be characterized with the notion of lower limit. 18. A function f : X → R on a topological space X is lower semicontinuous at some w ∈ X if and only if one has f (w) ≤ lim infx→w f (x).