MathPlotAI Logo MathPlotAI

Formal Continuity of Functions

Formal Continuity of Functions
Loading the math board and drawing, please wait…
2015-06-03 02:13
FunctionCalculusLimit
In collections Fun with Functions
This applet demonstrates the continuity of a function as x approaches a value. Key mathematical concepts include function continuity. Using GeoGebra commands such as Point, Element, If, Max, Textfield, etc., the function's graph is constructed, and its continuity at a specific point is observed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → SplitPoint
  2. Element(flist, nf) → g
  3. Point(xAxis) → goal
  4. If((xp ≥ 1) ∧ (xp < 2), 2, xp) → yx8
  5. If((xp * xm) ≤ 0, 0, Min[If[xp < 0, (-xp), If[xp ≤ 1, xp^(2), 1]], If[xm < 0, (-xm), If[xm ≤ 1, xm^(2), 1]]]) → ym3
  6. Max(If[xm < 0, (-xm), If[xm ≤ 1, xm^(2), 1]], If[xp < 0, 0, If[xp ≤ 1, xp^(2), 1]]) → yx3
  7. If(xm ≥ xSplit, y(Max[f_{right}, xm, xp]), If[xp < xSplit, y(Max[f_{left}, xm, xp]), Max[y(Max[f_{left}, xm, xSplit]), y(Max[f_{right}, xSplit, xp])]]) → yx5
  8. If(xm ≥ xSplit, y(Min[f_{right}, xm, xp]), If[xp < xSplit, y(Min[f_{left}, xm, xp]), Min[y(Min[f_{left}, xm, xSplit]), y(Min[f_{right}, xSplit, xp])]]) → ym5
  9. If((xm ≤ 1) ∧ (xp ≥ 1), Infinity, Max[f_7(xm), f_7(xp)]) → yx7
  10. If((xm ≤ 1) ∧ (xp ≥ 1), Infinity, Min[f_7(xm), f_7(xp)]) → ym7
  11. If((xm ≤ 0) ∧ (xp ≥ 0), NaN, y(Max[f_9, xm, xp])) → yx9
  12. Element(maxlist, nf) → ymax
  13. If((xm ≤ 0) ∧ (xp ≥ 0), NaN, y(Min[f_9, xm, xp])) → ym9
  14. Element(minlist, nf) → ymin
  15. Max(g(aa) - ymin, ymax - g(aa)) → epsilon
  16. Textfield(f_{left}) → inputBox1
  17. Textfield(f_{right}) → inputBox2
  18. OrthogonalLine(F, xAxis) → a
  19. OrthogonalLine(Xm, xAxis) → b
  20. OrthogonalLine(Xp, xAxis) → c
  21. OrthogonalLine(Ymin, yAxis) → d
  22. OrthogonalLine(Ymax, yAxis) → e
  23. Textfield(egoal) → inputBox3
  24. Polygon(G, C, B, A) → poly1, g_1, c_1, b_1, a_1

AI Drawing Prompt Reference

Draw a graph demonstrating function continuity. Show a piecewise-defined curve on a coordinate plane with a potential discontinuity at a split point. Include a draggable point on the x-axis for the split, and two input boxes for adjusting left and right function expressions. Add vertical dashed lines at sample positions on both sides of the split, and horizontal dashed lines indicating the minimum and maximum function values. Near the split, display a translucent rectangle representing the range of function values, with an adjustable error threshold. The layout is clean, suitable for high school calculus teaching.
The prompt above was compiled from the original drawing commands and is for reference only.