Differentiability of Real Functions

The Derivative

Definition: Derivative of a Real Function

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The derivative of $f$ in $x_0\in D$ is the limit

$$\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$$

More generally, the derivative of $f$ is the function which maps every $x\in D$ to $f^{\prime }(x)$.

NOTATION

$$f^{\prime }(x_0)\frac{\mathrm{d}f}{\mathrm{d}x}(x_0)\frac{\mathrm{d}}{\mathrm{d}x}f(x_0)$$

Definition: Higher-Order Derivatives

The $k$-th order derivative of $f$ is the derivative of the $(k-1)$-th order derivative of $f$:

• The $0$-th order derivative of $f$ is just $f$ itself;

• The first order derivative of $f$ is just the aforementioned derivative $f^{\prime }$;

• The second order derivative of $f$ is the derivative of $f^{\prime }$, etc.

NOTATION

$$f^{\prime }{f}^{\prime \prime }{f}^{\prime \prime \prime }{f}^{\mathrm{IV}}{f}^{\mathrm{V}}\cdots f^{(n)}$$ $$\frac{\mathrm{d}f}{\mathrm{d}x}\frac{{\mathrm{d}}^{2}f}{\mathrm{d}{x}^2}\frac{{\mathrm{d}}^{3}f}{\mathrm{d}{x}^3}\frac{{\mathrm{d}}^{4}f}{\mathrm{d}{x}^4}\frac{{\mathrm{d}}^{5}f}{\mathrm{d}{x}^5}\cdots \frac{{\mathrm{d}}^{n}f}{\mathrm{d}{x}^n}$$

Definition: Differentiability

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

We say that $f$ is differentiable at some $x\in D$ if the first-order derivative of $f$ in $x$ exists.

We say that $f$ is $k$-times (continuously) differentiable on $D$ if its $k$-th derivative exists (and is continuous).

Properties

Theorem: Differentiability $⟹$ Continuity

If $f:D\subseteq \mathbb{R}\in \mathbb{R}$ is differentiable at $x_0\in D$, then $f$ is also continuous at $x_0$.

PROOF

TODO

Mean Value Theorem for Derivatives

If $f:D\to \mathbb{R}$ is continuous on the closed interval $D=[a;b]\subset \mathbb{R}$ and differentiable on the open interval $(a;b)$, then there exists at least one $x_0\in (a;b)$ such that

$$f^{\prime }(x_0)=\frac{f(b)-f(a)}{b-a}$$

PROOF

TODO

Intuition: Geometric Meaning

The theorem says that there is at least one point $(x_0;f(x_0))$ on the graph of $f$, where the tangent line to $f$ is parallel to the secant line through the points $(a;f(a))$ and $(b;f(b))$.

TODO

Theorem: Darboux's Theorem

Let $f:D\to \mathbb{R}$ be a differentiable real function on a closed interval $D=[a;b]\subset \mathbb{R}$.

For each $\lambda \in \mathbb{R}$ such that $f^{\prime }(a)<\lambda <f^{\prime }(b)$ or $f^{\prime }(a)>\lambda >f^{\prime }(b)$, there exists some $c\in [a;b]$ such that

$$f^{\prime }(c)=\lambda$$

PROOF

TODO