Integrability & Differentiability of a Function: Implications for Derivatives

[bruno67]

A function f is both integrable and infinitely differentiable, i.e. $f\in L_1(\mathbb{R}) \cap C^{\infty}(\mathbb{R})$. Is it correct to say that this implies that the derivatives of f are also in $L_1(\mathbb{R})$? My reasoning: we have $I<\infty$, where

$$I=\int_{-\infty}^{\infty} f(x) dx = [x f(x)]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} xf'(x) dx = - \int_{-\infty}^{\infty} xf'(x) dx$$
where the boundary term disappears because, since f is integrable, we must have $f(x) =O(x^{-1-\alpha})$ for $|x|\to \infty$, for some $\alpha>0$. Hence $f'(x)=O(x^{-2-\alpha})$, and in general $f^{(n)}(x)=O(x^{-n-1-\alpha})$, for $|x|\to \infty$.

 

[micromass]

What about a function that oscillates really fast?? Its derivative will be very large.

 

[bruno67]

What about a function that oscillates really fast?? Its derivative will be very large.

Even in that case, I can't find an error with my proof.