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note: moved from mathoverflow, as off topic.

I'm currently reading this book: http://www.springer.com/us/book/9781441973993 And when speaking about germs of functions the only example provided of two functions having the same germ was a function and its Taylor expansion...

Is there any example of two functions (f,g) having same germ at point p without one being defined as a series converging to the other?

Indeed, the only examples I could think about where all such that:

$$ g(x) = \sum_{i=0}^{\infty} c_i b_i(x) $$

while

$$ \lim_{i \to \infty}\sum_{i} c_i b_i(x) = f(x)$$

for some appropriate basis of the appropriate Hilbert space ( so Taylor or Fourier Series for example ).

It seems a pretty useless definition to me! As g and f are basically the same function within the disc of convergence of the series. But I'm sure I'm missing the point.

If there is no such example then what is the point of this definition ?

user2346536
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  • You can remove the link to MO, as you have deleted your question there (which was a good action). – Silvia Ghinassi Nov 18 '15 at 15:35
  • Done. I thought people with high reputation score would still see it. Thanks for the info. – user2346536 Nov 18 '15 at 15:38
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    Yes, people with high reputation score on MO will, but that's not very relevant here. Anyway, a silly example: $f(x)=x$ and $g(x)=x$ for $x \in [-1,1]$, $g(x)=1$ for $x > 1$ and $g(x)=-1$ for $x<-1$. These functions surely have the same germ at $0$, for instance. Germs of function are meant to capture the behavior of functions locally, so you look only at the functions around a given point. – Silvia Ghinassi Nov 18 '15 at 15:41
  • Humm. I understand I guess... It is actually meant to just say "this two functions are actually the same and interchangeable within some neighborhood", so example will always involve this type of switch statement you did or a series. – user2346536 Nov 18 '15 at 15:46

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A silly example: $f(x)=x$ and $$g(x)= \begin{cases} x & x \in [-1,1]\\ 1 & x > 1 \\ -1 & x<-1. \end{cases} $$ These functions surely have the same germ at $0$, for instance. But also at any point $p \in (-1,1)$.

Germs of function are meant to capture the behavior of functions locally, so you look only at the functions around a given point. We care if the two functions coincide locally, but not globally.

Silvia Ghinassi
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