By R. C. Gunning
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Additional info for Lectures on complex analytic varieties: The local parametrization theorem
Example text
Clearly, f is monotonic as well as bimonotonic. Moreover, f (I × J) = [0, 2] is an interval in R. Note, however, that the real number 43 lies between f (0, 0) = 0 and f (1, 12 ) = 32 , but 43 is not 30 1 Vectors and Functions the value of f at any point on the 2-interval I(0,0),(1, 12 ) = [0, 1] × [0, 12 ]. Indeed, the image of this 2-interval is [0, 12 ] ∪ [1, 23 ], which is not an interval in R. 23 yields a characterization of the IVP. 25. Let D ⊆ R2 and let f : D → R be a function. Then for any 2-interval I × J ⊆ D, f has the IVP on I × J ⇐⇒ f (E) is an interval in R for every 2-interval E ⊆ I × J.
Xn , ym ), where m = 1, y0 = 0, and y1 = 1, we see that f would be of bounded variation. 10, f would have to be bounded, which is not the case. 15. y (0, 1) (1, 1) f (x, y) = 1 f (x, y) = 0 (0, 0) (1, 0) x Fig. 7. 19 (iii) and the points (xi , yi ) of the rectangle [0, 1] × [0, 1] that straddle the diagonal line y = 1 − x. (iii) Consider f : [0, 1] × [0, 1] → R defined by f (x, y) := 0 if x + y ≤ 1, 1 if x + y > 1. Then f is monotonically increasing in [0, 1] × [0, 1]. Indeed, given any (x1 , y1 ), (x2 , y2 ) ∈ [0, 1] × [0, 1] with (x1 , y1 ) ≤ (x2 , y2 ), we have x1 + y1 ≤ x2 + y2 , and hence x1 + y1 > 1 implies x2 + y2 > 1.
The tangent vectors to Γ1 , Γ2 , and Γ3 at the origin are (1, −1), (1, 1), and (2, 2), respectively. Hence Γ1 and Γ2 intersect transversally at (0, 0); also, Γ1 and Γ3 intersect transversally at (0, 0), but Γ2 and Γ3 do not intersect transversally at (0, 0). ) Let D ⊆ R2 and let (x0 , y0 ) be an interior point of D. We say that a function f : D → R has 28 1 Vectors and Functions 1. a local maximum at (x0 , y0 ) if there is δ > 0 such that Sδ (x0 , y0 ) ⊆ D and f (x, y) ≤ f (x0 , y0 ) for all (x, y) ∈ Sδ (x0 , y0 ), 2.