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.

(1)
\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:
(2)
\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:
(3)
\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}
(5)
\begin{align} \quad \int_a^b f'(x) \: dx \leq f(b) - f(a) \end{align}