Dynamic Demonstration of Continuity of a Function at a Point

① This example visually demonstrates the geometric meaning of continuity of a function at a point, showing that as the independent variable approaches the point, the function value approaches the function value at that point. ② The core mathematical concept is the epsilon-delta definition of continuity, involving limits, function values, neighborhoods, and identification of discontinuities. ③ Construction highlights: Using GeoGebra, functions h, f, g, points A, B, C, parameter a, slider n, and multiple text objects text1 to text9 along with buttons button1 and button2 are constructed. Sliders and buttons allow interactive movement of points, dynamically showing the approach of function values to limits, with auxiliary elements like cosine functions, parametric curves, and a rhombus. ④ Suitable for calculus teaching from high school to early university levels, for intuitive understanding of continuity and limits, and for classroom demonstrations or student self-exploration.