Continuous function whose integral is 0 but unbounded at inf

[passionate]

Can someone help me with this problem?

I need to construct a continuous function f: R into R such that its improper integral is zero, but which is unbounded as x approaches negative infinity and x approaches infinity.

 

[Deleted member 4993]

Re: Continuous function

Can someone help me with this problem?

I need to construct a continuous function f: R into R such that its improper integral is zero, but which is unbounded as x approaches negative infinity and x approaches infinity.

what about

\(\displaystyle f(x)\, = \, x^3\)

 

[pka]

Re: Continuous function

\(\displaystyle f(x)\, = \, x^3\)

The problem with that is \(\displaystyle \int\limits_{ - \infty }^\infty {x^3 dx}\) exists iff both \(\displaystyle \int\limits_{ - \infty }^0 {x^3 dx} \,\& \,\int\limits_0^\infty {x^3 dx}\) exist.
That is not the case.

But it is possible to construct triangles about each (n,0) with height n and base \(\displaystyle \frac{1}{2^{n-1}}\) so the sum of the of the areas is 2 on the positive side and –2 on
the negative side.

P.S.: Here is the construction.
Suppose that \(\displaystyle n \in Z^+\) then
\(\displaystyle f(x) = \left\{ {\begin{array}{rl} {n2^n \left( {x - n} \right) + n,} & {x \in \left[ {n - 2^{ - n} ,n} \right)} \\ { - n2^n \left( {x - n} \right) + n,} & {x \in \left[ {n,n + 2^{ - n} } \right)} \\ 0, & \mbox{else} \\\end{array}} \right.\)
\(\displaystyle x < 0 \Rightarrow \quad - f( - x)\)

 

[passionate]

Thanks a lot for your help. I really don't know how to tackle this problem. If something like this comes up on the exam, I don't think I'll be able to do it.