Relation between Continuity and differentiability

Relation between Continuity and Differentiability

Definitions:

  • Continuity: A function f(x) is continuous at a point c if f(c) is defined, the limit of f(x) as x approaches c exists, and these two values are equal.

  • Differentiability: A function f(x) is differentiable at a point c if the derivative f(c) exists.

Relationship:

  • Differentiability implies Continuity: If a function f(x) is differentiable at a point c, it must also be continuous at c.

    • Differentiability requires that the function is continuous and that the limit exists as h 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.

Key Concepts:

  • 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 x approaches a point.
  • Graphical Interpretation:

    • 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.