Thread: Infinitely many differenciable+compact support. Electromagnetism exercise

1. Infinitely many differenciable+compact support. Electromagnetism exercise

Verify that the function $f_a(x)=0$ if $|x|\geq |a|$ and $f_a (x)= exp \{ \frac{-1}{x+a}+ \frac{-1}{x-a} \}$ if $|x|< |a|$, $x \in \mathbb{R}$ is infinitely many times differentiable and has a compact support.

I have this exercise in an electromagnetism course and I never heard about compact support before. I've been told that if I can bound the function inside a sphere of a finite radius, then the function is of compact support. But reading the wikipedia article on it, I'm more lost than before.
Furthermore if I take the definition of a limit of a function, $f'_a(x)$ doesn't seem definied in $a$.
It might be some kind of distribution function and I never took any course on the subject so I'm really lost.
Any help is appreciated.

2. Originally Posted by arbolis
Verify that the function $f_a(x)=0$ if $|x|\geq |a|$ and $f_a (x)= exp \{ \frac{-1}{x+a}+ \frac{-1}{x-a} \}$ if $|x|< |a|$, $x \in \mathbb{R}$ is infinitely many times differentiable and has a compact support.

I have this exercise in an electromagnetism course and I never heard about compact support before. I've been told that if I can bound the function inside a sphere of a finite radius, then the function is of compact support. But reading the wikipedia article on it, I'm more lost than before.
Furthermore if I take the definition of a limit of a function, $f'_a(x)$ doesn't seem definied in $a$.
It might be some kind of distribution function and I never took any course on the subject so I'm really lost.
Any help is appreciated.
It's clearly differentiable except possibly at $x=\pm a$. For example it's clear that $\displaystyle \lim_{x\to a^-}\frac{f_a(x)f_a(a)}{x-a}=\lim_{x\to a^-}(x-a)^{-1}\exp\left(\frac{-1}{x+a}+\frac{-1}{x-a}\right)=0$ (since as $x\to a$ one has that $\frac{-1}{x-a}\to -\infty$ so that the exponential goes to zero faster than the polynomial) which agrees with $\lim_{x\to a^+}\frac{f_a(x)-f_a(a)}{x-a}=\lim_{x\to a^+}\frac{0}{x-a}=0$ and so $f_a(x)$ is differentiable at $a$. The exact same methodology shows that it is differentiablea at $-a$. Now, for $f_a$ to have compact support means that $\overline{f_a^{-1}\left(\mathbb{R}-\{0\}\right)}$ is compact. But, sine $\overline{f_a^{-1}\left(\mathbb{R}-\{0\}\right)}=[-a,a]$ the result is clear here.

3. Thanks a lot mathstud28. I did the calculation of the limit on my draft too fast... I looked only at the denominator and since it tends to 0 I thought the limit would be undefinied but as you say, the exponential tends to 0 faster than any polynomial and so the limit is well defined. Ok for the rest, I get it.