Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.