In "Numerical Mathematics" - Quarteroni the absolute condition number for a problem $$F(x,d)=0$$ where $d$ is the data and we want to find $x$, is defined by
$$K(d)=\sup_{\delta d\in D}\frac{\|\delta x\|}{\|\delta d\|}$$ where $D$ is the set of admissible pertubations.
There is no explanation for what the $\delta$ exactly stands for and I can only assume that $\delta d$ is a real number. The author then defines a resolution $G(d)$ which satisfies $F(G(d),d)=0$, meaning it is a solution.
I can only guess that $\delta x = G(d+\delta d)-G(d)$ but I have no idea.
An example is then discussed where $G(d)=\cos(d)-1$ for $d\in(-\pi/2,\pi/2)$
This should then yield $K(0)=\frac{2}{\pi}$ but I cannot reproduce this result.
In other books the condition number is defined differently, as $$K(d)={\displaystyle \lim _{\varepsilon \rightarrow 0}\sup _{\|\delta x\|\leq \varepsilon }{\frac {\|\delta f\|}{\|\delta x\|}}}$$ but this time $f$ is the $F$ in the notation above and $x$ is its argument.
I am completely confused by this and would need some clarification.