# Relation between Continuity and differentiability

## Relation between Continuity and Differentiability

### Definitions:

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

• Differentiability: A function $f\left(x\right)$ is differentiable at a point $c$ if the derivative ${f}^{\mathrm{\prime }}\left(c\right)$ exists.

### Relationship:

• Differentiability implies Continuity: If a function $f\left(x\right)$ 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.