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Continuity in Metric Spaces

Continuity in Metric Spaces
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2023-01-25 07:47
FunctionCalculus
This example demonstrates the concept of continuity in metric spaces. By constructing a series of points and functions, it illustrates the properties of continuous functions. Commands used include Sequence, Spline, Vector, Rotate, Translate, etc., suitable for high school and above students.

GeoGebra Commands & Parameters

  1. Sequence(Max[0, Min[1, k - s]], s, 0, top - 1) → sl
  2. Sequence(0 < sl(s) < 1, s, 1, top) → sc
  3. Sequence(0 < sl(s) ≤ 1, s, 1, top) → sc'
  4. Spline({A, B, C, D, E, F, G, H, I, J, K, L, A}, 3) → a
  5. Vector(A1, A2) → u
  6. Rotate(Translate[Dilate[a, 1.3], Vector[u - (0, 1)]], 1.12, A2) → b
  7. Spline({M, N, O, P, Q, R, S, T, U, V, W, Z, M}, 3) → c
  8. Rotate(Dilate[c, 1.3, A1], 1.2, A1) → d
  9. Translate(Rotate[Dilate[b, 1.5, A2], 3.13, A2], (0.2, -1)) → e
  10. Spline({A_1, B_1, C_1, D_1, E_1}, 3) → f
  11. Point(f) → F_1
  12. Point(f) → G_1
  13. Vector(F_1, G_1) → v
  14. Circle(A2, 2.25) → g
  15. PointIn(g) → H_1
  16. Circle(A2, H_1) → p
  17. Vector(A2, H_1) → w
  18. Circle(A1, (abs(w) * 1.1)) → h
  19. Point(h) → I_1
  20. Vector(A1, I_1) → i

AI Drawing Prompt Reference

Create a visual representation of continuity in metric spaces. Start by constructing a series of points and connecting them with cubic spline curves to form a function. Then, generate multiple transformed versions of the spline using translation, dilation, and rotation, with vectors guiding the transformations. Finally, add circles and points to highlight continuity, where the radius of the circles dynamically changes based on vector lengths, and points are located on the circles relative to specific vectors.
The prompt above was compiled from the original drawing commands and is for reference only.