Mathematics
Mathematics
Problem
Which of the following statements correctly describes the relationship between continuity and differentiability of a function at a point ? a) If is continuous at , then must be differentiable at . b) If is differentiable at , then must be continuous at . c) If is not continuous at , then can still be differentiable at . d) Continuity and differentiability are independent; neither property implies the other at .Intuitive explanation
Differentiability is the stronger condition: if a function is differentiable at , it must be continuous there, but continuity alone doesn't guarantee differentiability. The key insight is that differentiability requires a smooth, well-defined slope, while continuity only requires the function to have no jumps or breaks—a function can be continuous but have a sharp corner or cusp where the derivative doesn't exist.
Intuitive explanation
Differentiability is the stronger condition: if a function is differentiable at , it must be continuous there, but continuity alone doesn't guarantee differentiability. The key insight is that differentiability requires a smooth, well-defined slope, while continuity only requires the function to have no jumps or breaks—a function can be continuous but have a sharp corner or cusp where the derivative doesn't exist.