Continuity and Differentiability

Imagine constructing a physical bridge spanning a deep gorge. For a vehicle to successfully cross, the physical path must be completely unbroken—any missing segment means catastrophe. Yet, merely connecting the road is not sufficient for a safe journey. If the bridge sections meet at a harsh, jagged angle, the wheels will abruptly change direction, subjecting the vehicle to immense, infinite forces that will shatter its suspension. In calculus, these two physical necessities—connectedness and smoothness—are mathematically formalized as continuity and differentiability. Understanding the precise relationship between these two properties is central to mastering single-variable calculus.

As mathematics educators, you will find that students intuitively understand continuity as "drawing a graph without lifting your pencil." While this heuristic is a helpful starting point, it spectacularly fails when dealing with rigorous analytical definitions or when a graphing calculator's pixelated screen obscures a single undefined point. To prepare your students—and to master the Praxis 5165 exam yourself—we must transition from geometric intuition to strict algebraic conditions.