Differentiability and Partial Derivatives

Statements

Proposition. Let $U\subset {\mathbb{R}}^n$ be an open set, if the vector-valued function $f:U\to {\mathbb{R}}^m$ is differentiable at $a\in U$, then it is continuous at $a$ and all the partial derivatives exist at $a$. Moreover, $R(x,a)$ and $A$ exist and $A$ is the Jacobian matrix and Equation (differentiable) is given by

$$\begin{array}{c}F(x)=F(a)+DF(a)(x-a)+||x-a||R(x,a)\end{array}$$

When $f$ is a function, the coefficients $b_i$ are uniquely determined for each $a$ which $f$ is differentiable; in fact $b_i={\frac{\partial f}{\partial x^i}|}_a.$ Conversely, if the partial derivatives of $f$ at $a$ exist and are continuous, then $f$ is differentiable at $a$.

References

@Boo75 Chapter 2, Proposition 1.1