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}