Continuity versus Darboux Property for Real Functions

This GeoGebra example demonstrates the difference and relationship between continuity and the Darboux (intermediate value) property of a real function defined on an interval. The core mathematical concepts covered include real functions, intervals, continuity, the Darboux property, the intermediate value theorem, limits, and function graphs. Construction highlights: it employs GeoGebra commands such as If, Min, Max, Polygon, Segment, Line, and Intersect, along with objects including f, f_l, f_r, a, b, y_1, y_2, P, Q, R, S, poly1, and related variables, to construct piecewise function graphs, mark endpoints and extrema, draw segments and polygons, and compute intersection points. It is suitable for senior high school, undergraduate mathematical analysis, or advanced algebra courses, as well as classroom demonstrations and self-study, helping learners grasp that continuity implies the Darboux property, while the converse does not necessarily hold.