Continuity and Differentiability Explained (using Limits) with Examples and Proof

In this Calculus tutorial, we break down the key ideas of continuity and differentiability using limits. We start by reviewing the limit definition of the derivative and the formal definition of continuity, then explore how these concepts are connected. From there, we dive into an important result: every differentiable function is continuous—but not every continuous function is differentiable. To make this clear, we analyze the absolute value function, showing how it satisfies the definition of continuity while failing the limit definition of the derivative at a critical point. We wrap up by proving that differentiability implies continuity, giving you both the intuition and the formal reasoning behind this foundational concept. Perfect for students looking to strengthen their understanding of limits, derivatives, and the deeper connections in Calculus. The Limit Definition of Continuity: https://youtu.be/z-C0Oell-LU The Limit Definition of a Derivative: https://youtu.be/ATFu_j42xrk Thank You for your Support! Patreon: https://www.patreon.com/cw/LinzeyGMath For Practice: https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/e/differentiability-at-a-point-graphical #learnmath #practicemaths #maths #teachmath