Flat tail?

**davidmccormick** · April 16th 2010, 12:12 PM

Suppose $f:\mathbb{R} \longrightarrow [0,\infty)$ is continuous and that $\int f < \infty$ . Is it always true that $\lim_{x\to\infty}f(x)=0?$

**Drexel28** · April 16th 2010, 12:17 PM

Originally Posted by davidmccormick

Suppose $f:\mathbb{R} \longrightarrow [0,\infty)$ is continuous and that $\int f < \infty$ . Is it always true that $\lim_{x\to\infty}f(x)=0?$

Clearly we must have that eventually $f(x)$ is non-increasing. So, let $f(x)$ be non-increasing for $x\geqslant N\in\mathbb{N}$ . Then, $f$ is continuous non-negative non-increasing function. Thus, $\int_{N}^{\infty}f<\infty\implies \sum_{n=N}^{\infty}f(n)<\infty$ and so $\lim_{n\to\infty}f(n)=\lim_{x\to\infty}f(x)=0$ .

**Laurent** · April 16th 2010, 01:56 PM

Originally Posted by Drexel28

Clearly we must have that eventually $f(x)$ is non-increasing.

What do you mean?

In general, functions $f\geq 0$ such that $\int f<\infty$ are not non-increasing. Some of them converge to 0, others don't.

Here's an example. Let me describe the graph of $f$ . It consists of triangular peaks that get thinner and higher : around $n$ , we put a triangular peak of width $\frac{1}{n^3}$ and height $n$ . (Is it clear?)

The area of the triangle at $n$ is $\frac{1}{n^2}$ . Since $\sum_n \frac{1}{n^2}<\infty$ , the total area under the curve is finite, i.e. $\int f(x)dx<\infty$ . However, $f$ clearly does not converge to 0. We even have $f(n)=n\to +\infty$ for $n\in\mathbb{N},n\to\infty$ .

**Drexel28** · April 16th 2010, 02:00 PM

Originally Posted by Laurent

What do you mean?

In general, functions $f\geq 0$ such that $\int f<\infty$ are not non-increasing. Some of them converge to 0, others don't.

Here's an example. Let me describe the graph of $f$ . It consists of triangular peaks that get thinner and higher : around $n$ , we put a triangular peak of width $\frac{1}{n^3}$ and height $n$ . (Is it clear?)

The area of the triangle at $n$ is $\frac{1}{n^2}$ . Since $\sum_n \frac{1}{n^2}<\infty$ , the total area under the curve is finite, i.e. $\int f(x)dx<\infty$ . However, $f$ clearly does not converge to 0. We even have $f(n)=n\to +\infty$ for $n\in\mathbb{N},n\to\infty$ .

I misread the question

...I gave myself extra conditions I didn't have.

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