Differentiability at Discontinuity?

📌 CONCEPT: A function is said to be differentiable at a point if it is continuous at that point and its derivative exists at that point.

📐 RULE / FORMULA: For a function f(x) to be differentiable at a point x = a, the function must be continuous at x = a and the limit of the difference quotient must exist at x = a.

💡 WORKED EXAMPLE: Consider f(x) = |x| at x = 0. To check differentiability, we first check continuity. Since lim x→0 |x| = 0 = f(0), the function is continuous at x = 0. Now, we check the existence of the derivative. The derivative of f(x) = |x| does not exist at x = 0 because the left and right derivatives are not equal. Hence, f(x) = |x| is not differentiable at x = 0.

⚠️ COMMON MISTAKE: Students often assume that a function is differentiable at a point if it is continuous at that point, but this is not true. Differentiability requires the existence of the derivative, not just continuity.