Critical Points Matter?

📌 CONCEPT: A function f(x) can be differentiable at a point x=c, even if its derivative f'(x) is not defined at the same point, as long as f(x) is continuous at x=c. This means that the function can have a critical point where it is not differentiable, but still be continuous at that point.

📐 RULE / FORMULA: If a function f(x) is continuous at a point x=c, but its derivative f'(x) is not defined at the same point, it is not necessarily a contradiction to the differentiability of f(x) at x=c.

💡 WORKED EXAMPLE: Consider the function f(x) = x^3 at the point x=0. It is continuous at x=0, but its derivative f'(x) = 3x^2 is not defined at x=0. However, f(x) is still differentiable at x=0.

⚠️ COMMON MISTAKE: Many students assume that if a function is continuous at a point, it must also be differentiable at that point, which is not necessarily true.