Differentiability

Definition

f is differentiable at a, if:

1. f(x) is continuous at a;
2. 

Alternative definition:
The function is differentiable at , if there exists an m × n matrix Df(x) such that:

Matrix Df(x) is the derivative of f at x.

If f is differentiable at all points , then f is said to be differentiable.

Properties of differentiability

(1) f is continuous at

Note

• At kink, the function is continuous but not differentiable.
• differentiable continuous, but not verse visa.

Example