An application of Fatou's lemma and a differential function 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)?
 A: 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
$$
  \begin{split}
    \int_{-1}^1
      n \cdot |f(x + 1/n) - f(x)|
    \, d x
    & = \int_{-1}^1
          n \cdot \bigg| \int_x^{x + 1/n} f'(t) \, d t \bigg|
        \, d x \\
    & \leq \int_{-1}^1
             n \cdot \int_{\mathbb{R}} 1_{x \leq t \leq x + 1/n} |f'(t)| \, d t
           \, d x \\
    & \leq n
           \int_{-1}^{1 + 1/n}
             |f'(t)|
             \int_{\mathbb{R}}
               1_{t - 1/n \leq x \leq t}
             \, d x
           \, d t \\
    & \leq n \int_{-1}^{1 + \delta}
               |f'(t)| \cdot \frac{1}{n}
             \, d t
      =    \int_{-1}^{1+\delta} |f'(t)| \, d t .
  \end{split}
$$
