Suppose I have the following conditions:
$f(0)=a$
$f'(0)=b$
$f''(0)=c$
...
$f^{\infty'}(0)= $ something
Is there an unique $f(x)$ which satisfies this?
In other words: given the values of some function and all of its derivatives at point $a$, and given the function is infinitely differentiable, only one function exists that satisfies them?
No. There are $C^\infty$ "bump functions".
To get uniqueness, you'll want to assume the function is analytic.