Differentiability vs Continuity

AP Calculus · Axiom Academy

LESSON Differentiability vs Continuity Understanding when derivatives exist A function is continuous at a point if there are no breaks, jumps, or holes in its graph at that point. A function f is continuous at x = a if: Left: A continuous function with no breaks. Right: A discontinuous function with a jump. Step 2: What is Differentiability? A function is differentiable at a point if the derivative exists at that point. This requires more than just continuity—the function must be smooth with no sharp corners or cusps. A function f is differentiable at x = a if exists and is finite. If f is differentiable at a, then f is continuous at a. If f is continuous at a, then f is differentiable at a. In other words: Differentiability is a stronger condition than continuity. Step 4: Where Derivatives Don't Exist Even if a function is continuous, the derivative fails to exist at: The left and right derivative limits are different. The slope becomes undefined/infinite. Example: f(x) = x^(2/3) at x = 0 The tangent line is vertical (infinite slope). Example: f(x) = x^(1/3) at x = 0 Jump, infinite, or removable discontinuity. You've learned the difference between continuity and differentiability! Next: The Power Rule—the fastest way to find derivatives.

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