Carr & Madan (1999): Proof that any Payoff can be replicated using Bonds and Options

Question

I am reading through "Towards a Theory of Volatility Trading" by Carr & Madan (1999) and I'm pretty sure they make a mistake in their key proof (Appendix 1). They write:

$f(K)\mathcal{1}(F<K)|^{k}_{0} - \int^{k}_{0}f'(K)\mathcal{1}(F<K)dK + f(K)\mathcal{1}(F\geq{K})|^{\infty}_{k} - \int^{\infty}_{k}f'(K)\mathcal{1}(F\geq{K})dK = f(k) - \int^{k}_{0}f'(K)\mathcal{1}(F<K)dK - \int^{\infty}_{k}f'(K)\mathcal{1}(F\geq{K})dK$

I just don't understand how they obtain that $f(K)\mathcal{1}(F<K)|^{k}_{0} + f(K)\mathcal{1}(F\geq{K})|^{\infty}_{k} = f(k)$.... Can anyone please help me understand this?

Thanks!

Answer 1

OK no sweat - I just found a more detailed proof here:

http://www.frouah.com/finance%20notes/Payoff%20function%20decomposition.pdf

thank you nonetheless Masacroso