Relation between Continuity and differentiability
Relation between Continuity and Differentiability
Continuity: A function is continuous at a point if is defined, the limit of as approaches exists, and these two values are equal.
Differentiability: A function is differentiable at a point if the derivative exists.
Differentiability implies Continuity: If a function is differentiable at a point , it must also be continuous at .
- Differentiability requires that the function is continuous and that the limit exists as approaches 0 in the difference quotient for the derivative.
Continuity does not always imply Differentiability: A function can be continuous at a point where it's not differentiable.
- Discontinuities like sharp corners, cusps, or vertical tangents may lead to continuity without differentiability.
Conditions for Differentiability and Continuity:
- Differentiability includes the existence of a limit defining the derivative.
- Continuity involves ensuring the function's value matches the limit as approaches a point.
- A function can be continuous without having a well-defined tangent at that point (non-differentiable).
- Differentiability implies smoothness without abrupt changes or sharp corners.