Pointwise Convergence Does Not Preserve Continuity: A Counterexample

① This example constructs a sequence of functions to visually demonstrate that the pointwise limit of continuous functions may be discontinuous, showing that continuity is not preserved under pointwise convergence. ② The core mathematical concepts involve pointwise convergence of function sequences, continuity, and properties of limit functions, interacting with real sequences and function limits. ③ Construction highlights: using GeoGebra commands Segment, Semicircle, and Sequence to generate a series of function graphs, with text objects (text7, text5, text3) labeling key points and explanations; objects include C, g, D, E, c, b, f, nn, l1, and a slider or sequence to animate the convergence process. ④ Suitable for university-level mathematical analysis or calculus courses when teaching convergence properties of function sequences, and also for advanced high school mathematics enrichment to help students intuitively grasp the difference between pointwise and uniform convergence.