# If $f\in L^1(\mathbb R)$ and $f'\in L^1(\mathbb R)$, then $\lim_{x\to \infty }f(x)=0$.

Let $$f,f'\in L^1(\mathbb R)$$. Prove that $$\lim_{x\to \infty }f(x)=0.$$

First of all, is $$f'$$ defined a.e. ? Because there are no assumption on the fact that $$f$$ is derivable. So, is $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ for almost every $$x$$ ?

My attempt for the statement : Suppose WLOG $$f'>0$$. I'm not sure if it's true, but I would say that : Let $$\varepsilon >0$$. Since $$f'$$ is $$L^1$$, there is a ball $$[a,b]$$ s.t. $$\int_{]-\infty ,a[\cup]b,+\infty [}f'(x)dx<\varepsilon .$$ In particular, if $$x>y>b$$, then $$f(x)-f(y)=\int_y^x f'(t)dt<\varepsilon,$$ but I just can conclude that $$\lim_{x\to \infty }f(x)-f(y)<\varepsilon .$$

• For the first question, why not? It's written "$f' \in L^1(\Bbb{R})$" in the title. If $f'$ is not defined a.e., this condition would be meaningless. It suffices to show the case for $f\ge0$. Try a proof by contradiction. – GNUSupporter 8964民主女神 地下教會 May 18 at 8:34
• @GNUSupporter8964民主女神地下教會: Thanks but how do you think $f'$ is defined ? In the weak sense or as I defined in my post ? – user659895 May 18 at 8:40
• @user659895 Do you know that there exist continuous strictly increasing functions $f$ with $f'=0$ almost everywhere. Fundamental Theorem of Calculus (FTC) is proved under the assumption that $f$ has a continuous derivative and you cannot make such assumptions in this question. So use of FTC is not at all admissible here. – Kavi Rama Murthy May 18 at 23:15

I think the question is not properly posed. Normally the statement is interpreted as follows: if $$f$$ is integrable and differentiable almost everywhere with $$f'$$ also integrable then $$f(x) \to 0$$ as $$x \to \infty$$. This statement is false. $$f(x)=1$$ when $$x$$ is an integer and $$0$$ otherwise gives a counterexample. Some additional assumptions on $$f$$ are necessary to prove that $$f(x) \to 0$$ as $$x \to \infty$$.

• but $f=1$ is not integrable... – user659895 May 18 at 12:04
• $f=1$ only at integer points which form a set of measure $0$. @user659895 – Kavi Rama Murthy May 18 at 12:05
• Strange because to prove that the fourier transform of the derivative is $\mathcal F(f')(\alpha )=-2i\pi \alpha \mathcal F(f)(\alpha )$, in my exercise the only assumption are $f,f'\in L^1$ – user659895 May 18 at 12:07
• In such results it is generally assumed that $f$ is an absolutely continuous function such that $f$ and $f'$ are integrable. – Kavi Rama Murthy May 18 at 12:09
• It appears that the downvoter doesn't care about Mathematical rigor. – Kavi Rama Murthy May 18 at 12:19

The idea is to show that $$f(x)-f(0)=\int_0^x f'$$ hence $$f$$ has a limit in $$\infty$$, which must be $$0$$ because $$f\in L^1$$.

On the one hand, observe that exists $$l = \lim_{x \to \infty} f(x)$$. It is because for each $$x \in \mathbb{R}$$ $$f(x) - f(a) = \int_{(a , x)} f' \quad \Longrightarrow \quad \lim_{x \to \infty} f(x) = f(a) + \int_{(a , \infty)} f' \in \mathbb{R}\mbox{,}$$ as $$f' \in L^1(\mathbb{R})$$. On the other hand, if $$l \neq 0$$, then there exist $$\varepsilon , x_0 \in (0 , \infty)$$ such that $$|f(x)| \geq \varepsilon$$ for all $$x \in \mathbb{R}$$ with $$x \geq x_0$$. Therefore $$\int_{\mathbb{R}} |f| \geq \int_{[x_0 , \infty)} |f| \geq \varepsilon \int_{[x_0 , \infty)} 1 = \infty\mbox{,}$$ which is a contradiction, as $$f \in L^1(\mathbb{R})$$.

• You are making some strong assumptions. The hypothesis does not guarantee that the Fundamental Theorem of Calculus is applicable. – Kavi Rama Murthy May 18 at 11:54
• Hypothesis guarantee that the FTC is applicable because $f$ is continuous on $\mathbb{R}$, you can remember the FTC's hypothesis here: en.wikipedia.org/wiki/Fundamental_theorem_of_calculus – joseabp91 May 18 at 12:02
• It is not given that $f$ is continuous and it is not given that $f$ is differentiabel at every point. The statement, as it stands, is false as I have pointed out in my answer. – Kavi Rama Murthy May 18 at 12:04
• I think that you are wrong: if $f \in L^1(X)$, normally we suppose that $f : X \to \mathbb{R}$. – joseabp91 May 18 at 12:11
• In analysis we often talk of $f'$ even when $f$ need not be everywhere-differentiable, only requiring that $f$ be differentiable almost-everywhere instead. Under this interpretation, FToC need not hold. Even if $f$ is everywhere differentiable and $f'$ is integrable, it is a quite non-trivial result that FToC holds for this case, which is often skipped in many analysis textbook. (Rudin's RCA is a notable exception, and he proves this fact using Vitali–Carathéodory theorem.) – Sangchul Lee May 19 at 7:00