Lebesgue's Theorem, Differentiability of Monotone Functions Review
Lebesgue's Theorem and Differentiability of Monotone Functions Review
We will now review some of the recent material regarding Lebesgue's theorem and differentiability of monotone functions.
- On the Upper and Lower Derivatives of Real-Valued Functions page we said that if $f$ is a real-valued function defined on an open interval $I$ then the Upper Derivative of $f$ at $x$ is defined as:
\begin{align} \quad \overline{D} f(x) = \limsup_{h \to 0} \frac{f(x + h) - f(x)}{h} \end{align}
- And the Lower Derivative of $f$ at $x$ is defined as:
\begin{align} \quad \underline{D} f(x) = \liminf_{h \to 0} \frac{f(x + h) - f(x)}{h} \end{align}
- We said that $f$ is Differentiable at $x$ if the upper and lower derivatives of $f$ at $x$ are finite and equal.
- We then proved an important result regarding increasing functions $f$ and the upper derivative of $f$. We proved that if $f$ is an increasing function on $[a, b]$ then for all $\alpha \in \mathbb{R}$ with $\alpha > 0$ then:
\begin{align} \quad m^* ( \{ x \in (a, b) : \overline{D} f(x) \geq \alpha \}) \leq \frac{1}{\alpha} [f(b) - f(a)] \end{align}
(4)
\begin{align} \quad m^* ( \{ x \in (a, b) : \overline{D} f(x) = \infty \}) = \infty \end{align}
- On the Lebesgue's Theorem for the Differentiability of Monotone Functions page we proved a very important result. We proved that if $f$ is a monotone function on $[a, b]$ then $f$ is differentiable almost everywhere on $(a, b)$.
- On the The Lebesgue Integral of the Derivative of an Increasing Function we proved another important result. We proved that if $f$ is increasing on $[a, b]$ then $f'$ is nonnegative where it is defined as furthermore:
\begin{align} \quad \int_a^b f'(x) \: dx \leq f(b) - f(a) \end{align}