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 ${f}^{\mathrm{\prime}}(c)$ 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 welldefined tangent at that point (nondifferentiable).
 Differentiability implies smoothness without abrupt changes or sharp corners.