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Exploration of Non-uniform Continuity of 1/x on (0,1]

Exploration of Non-uniform Continuity of 1/x on (0,1]
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2023-08-05 16:28
FunctionCalculusLimitGraphing
A GeoGebra visualization exploring non-uniform continuity of f(x)=1/x on the interval (0,1]. This example demonstrates the ε-δ definition of continuity and shows why f(x)=1/x fails to be uniformly continuous on this bounded interval despite being continuous at every point. Key commands include Point, Polygon, Intersect, Segment, and RemoveUndefined to handle the domain boundary. The construction visualizes ε-δ rectangles and dynamically shows that no single δ works uniformly for all points as x approaches 0. Suitable for calculus and analysis students learning about limits and continuity concepts.

GeoGebra Commands & Parameters

  1. Point(Segment[(0, 0), (2, 0)]) → P
  2. Polygon(E, C, B, D) → q1, e, c, b, d
  3. Intersect(f, e, (0.8594027451992855, 1.163598796473604)) → F
  4. Intersect(f, b, (1.2594027451992855, 0.7940271718573727)) → G
  5. RemoveUndefined({F, G}) → int
  6. Length(int) → lengthInt
  7. Segment(A, P) → g
  8. Segment(A, H) → h
  9. Segment(I, H) → i
  10. Segment(H, J) → j
  11. Segment(L, P) → k
  12. Segment(P, K) → l

AI Drawing Prompt Reference

Draw the graph of f(x)=1/x on the interval (0,1] to illustrate its continuity but non-uniform continuity. Highlight the behavior as x approaches 0, where the function tends to infinity, and as x approaches 1, where it tends to 1. Use curves and marked points to visualize these changes, helping to understand the δ issue in the ε-δ definition.
The prompt above was compiled from the original drawing commands and is for reference only.