An application of Fatou's lemma and a differential function

Question

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a differentiable function (the derivative is not always continuous). Let $f$ satisfy the following condition. \begin{equation} \sup_{n \in \mathbb{N}} \int_{-1}^1 n|f(x+1/n)-f(x)|\,dx<\infty.\tag{A} \end{equation} Then, the Fatou's lemma implies that \begin{equation} \int_{-1}^1|f'(x)|\,dx<\infty. \end{equation} Condition (A) holds if $f$ is Lipschitz continuous. Is there a differentiable function that is not Lipschitz continuous but satisfies condition (A)?

Answer 1

Contrary to what the other answer claims, it is not true that $f$ needs to be Lipschitz continuous. Rather, it is sufficient that $f$ be locally absolutely continuous and such that $$ \int_{-1}^{1+\delta} |f'(t)| \, d t < \infty \quad \text{for some } \delta > 0. \tag{$\ast$} $$ The proof is below. One can probably even remove the "$+ \delta$" part, but I was too lazy to do so. As an explicit example of a function which satisfies this condition but which is not Lipschitz continuous, consider $$ f(x) = \begin{cases} 0, & \text{if } x < 0, \\ \sqrt{x} , & \text{if } 0 \leq x < 1, \\ 1 , & \text{if } x \geq 1. \end{cases} $$

Now, let us prove that condition $(\ast)$ is indeed sufficient. To see this, note for all $n \in \mathbb{N}$ with $1/n < \delta$ that $$ \begin{split} \int_{-1}^1 n \cdot |f(x + 1/n) - f(x)| \, d x & = \int_{-1}^1 n \cdot \bigg| \int_x^{x + 1/n} f'(t) \, d t \bigg| \, d x \\ & \leq \int_{-1}^1 n \cdot \int_{\mathbb{R}} 1_{x \leq t \leq x + 1/n} |f'(t)| \, d t \, d x \\ & \leq n \int_{-1}^{1 + 1/n} |f'(t)| \int_{\mathbb{R}} 1_{t - 1/n \leq x \leq t} \, d x \, d t \\ & \leq n \int_{-1}^{1 + \delta} |f'(t)| \cdot \frac{1}{n} \, d t = \int_{-1}^{1+\delta} |f'(t)| \, d t . \end{split} $$