Intereting Posts

Rational solutions to $a+b+c=abc=6$
Distribution of Brownian motion
Irreducible representations (over $\mathbb{C}$) of dihedral groups
Integers that satisfy $a^3= b^2 + 4$
Double integral of an function odd with respect to $y$ over a domain symmetric with respect to $x$ axis
How to use Fermat's little theorem to find $50^{50}\pmod{13}$?
Extensions and contractions of prime ideals under integral extensions
Probability of opening all piggy banks
$e^{\left(\pi^{(e^\pi)}\right)}\;$ or $\;\pi^{\left(e^{(\pi^e)}\right)}$. Which one is greater than the other?
Some pecular fractional integrals/derivatives of the natural logarithm
Find a way from 2011 to 2 in four steps using a special movement
Determining the Asymptotic Order of Growth of the Generalized Harmonic Function?
Why is this weaker then Uniform Integrability?
Is every linear ordered set normal in its order topology?
Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$

Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?

I know there are such functions, but I just can’t think of any example.

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- Evaluating $\int_{0}^{\infty} \left ^{-1}\mathrm{d}x$
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- $e$ as the limit of a sequence
- Is supremum over a compact domain of separately continuous function continuous?

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- Integral of $\int \frac {\sqrt {x^2 - 4}}{x} dx$
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- Calculus book recommendations (for complete beginner)
- Integral $\int_0^\infty\sin{(x^4)} dx$
- Is there any integral for the Golden Ratio?
- Method of Steepest Descent and Lagrange
- Why is the empty set a subset of every set?

Here is a picture (not very accurate, I know), to see how to construct a counter-example:

$\qquad\qquad\qquad$

The $n$-th triangle centered at $x=n$ have basis of length $1/n^2$.

This is Friedrich Philipp’s idea.

Let

$$

f(x)=\begin{cases}n^2(x-n),&\ x\in[n,n+1/n^2], \\ -n^2x+n^3+2,&\ x\in[n+1/n^2,n+2/n^2]\\ 0,&\ x\in[n+2/n^2,n+1)

\end{cases}

$$

Then $f$ is continuous, $f(x)\geq0$ for all $x$, and

$$

\int_0^\infty f(x)\,dx=\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6.

$$

Note also that, by pushing this idea, we can get $f$ to be unbounded (by making the triangles thin quicker and get higher).

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