Differentiable Function

analysis

Definition

Differentiable Function

A function $f:D\to \mathbb{R}$ is called differentiable in point $x_0$, if the limit ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$. The limit is called the derivative of $f$ at point $x_0$ and is named $f^{\prime }(x_0)$.

A function $f(x)$ is called $n$-times differentiable in point $x_0$, if the $n$-th derivative $f^{(n)}(x_0)$ exists, which can be defined recursively:

$$f^{(n)}(x)=\frac{d}{dx}{f}^{(n-1)}(x),f^{(1)}(x)=f^{\prime }(x)$$

If $f^{(n)}$ is also continuous in $x_0$, then $f(x)$ is called $n$-times continuously differentiable in $x_0$.

Continuity

Differentiability Implies Continuity

If a function $f$ is differentiable in $x_0$, then $f$ is also continuous at $x_0$.

Differentiability Implies Continuity

Let $f(x)$ in $x_0$ be differentiable. Thus:

$$\begin{array}{rl}\underset{x\to x_0}{\lim }\left(f(x)-f(x_0)\right)& =\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)\\ & =\underset{f^{\prime }(x)}{\underbrace{\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}}}\cdot \underset{0}{\underbrace{\underset{x\to x_0}{\lim }(x-x_0)}}\\ & =0\end{array}$$

Linear Approximation

A function $f(x)$ is differentiable in $x_0$ if and only if it is linearly approximatable:

$$f(x)-f(x_0)=f^{\prime }(x_0)(x-x_0)+R(x)$$

with $R(x)=o(x-x_0)$ (where $o(\dots )$ refers to the Big O Notation).