Let $I \subset \mathbb{R}$ be a nondegenerate interval, let $E \subset I$ be finite, and let $f: I \to I$ be a Borel measurable map such that for every $y \in I \setminus E$ there exists an open interval $U_y \subset \mathbb{R}$ with $y \in U_y$ and a $C^1$ map $F_y: U_y \to \mathbb{R}$ satisfying $F_y = f$ on $U_y \cap I$. Let $\mu$ be an ergodic $f$-invariant probability measure on the Borel $\sigma$-algebra of $I$. Assume that the observable $t \mapsto \log |f'(t)|$, defined on $I \setminus E$ using these local $C^1$ representatives and extended arbitrarily to $E$, belongs to $L^1(I,\mu)$, and assume that

\begin{align*} \mu\left(\left\{x \in I : f^j(x) \notin E \text{ for every } j \in \{0,1,2,\dots\}\right\}\right)=1. \end{align*}

Then for $\mu$-almost every $x \in I$, the derivative $(f^n)'(x)$ exists for every $n \in \mathbb{N}$ and

\begin{align*} \lambda(x) := \lim_{n \to \infty} \frac{1}{n}\log |(f^n)'(x)| = \int_I \log |f'(t)|\, d\mu(t). \end{align*}