Differentiability

Differentiable

Scaler-Valued Functions

Let $f:X\mapsto \mathbb{R}$ be a function where $X\in\mathbb{R}^n$ is open, suppose $f_{x_j}(\mathbf{a})$ exists for $j=1\dots n$ where $\mathbf{a}\in X$. Define

$$h(\mathbf{x})=f(\mathbf{a})+f_{x_1}(\mathbf{a})(x_1-a_1)+f_{x_2}(\mathbf{a})(x_2-a_2)+\dots+f_{x_n}(\mathbf{a})(x_n-a_n)$$

i.e., $h(\mathbf{x})$ is the first order Taylor expension of $f(\mathbf{x})$.

We say that $f$ is differentiable at $\mathbf{a}$ if

$$\lim_{\mathbf{x\to a}}\frac{f(\mathbf{x})-h(\mathbf{a})}{\Vert \mathbf{x}-\mathbf{a} \Vert}=0$$

Moreover, we say $f$ is differentiable if it is differentiable at every point in $X$.

Proposition: Let $a$ be an interior point, if $f$ is differential at $a$, then $f$ is continuous at $a$.

Vector-Valued Functions

Define $\boldsymbol{h}()$ to be the 1^st order Taylor expension of $\boldsymbol{f}()$ near $\boldsymbol{a}$. Since it’s multi-variable, we use Jacobian Matrix

$$\boldsymbol{h}(\boldsymbol{x})=\boldsymbol{f}(\boldsymbol{a}) + D \boldsymbol{f}(\boldsymbol{a})(\boldsymbol{x}-\boldsymbol{a})$$

We say $\boldsymbol{f}$ is differentiable at $\boldsymbol{a}$ if

$$\lim_{\boldsymbol{x}\to \boldsymbol{a}} \frac{\Vert\boldsymbol{f}(\boldsymbol{x})-\boldsymbol{h}(\boldsymbol{x})\Vert}{\Vert \boldsymbol{x}-\boldsymbol{a} \Vert}=0$$