Bouncing back and forth between anti-difference and finite difference in finite calculus at exponential functions.

Question

I've been reading 'concrete mathematics(knuth)' and just don't get how I'm supposed to bounce back and forth between anti-difference and finite difference in finite calculus, specifically at exponential functions. I really have nowhere to ask about this. Please help.$$$$ I get that if $f(x)=c^x$ its finite difference is $$\Delta f(x)=f(x+1)-f(x)=c^{x+1}-c^x=(c-1)c^x.$$ And I also get $$\Delta^{-1}f(x)=\frac{c^x}{c-1}$$ since $$F(x+1)-F(x)=\frac{c^{x+1}-c^x}{c-1}=c^x.$$ But what I don't get is how to get finite differece of the exponential function through anti-difference to getback to the original exponential function $c^x$. Please tell me how this is done. And one other thing. I understand why the anti-difference of $c^x$ is $\frac{c^x}{c-1}$ by itself. But I don't understand how it came out from the finite difference $(c-1)c^x$. It seems that there is pretty simple and straightforward reason as to why that is, since there's no explanation about it. But I just don't see it. $$\sum{\Delta{f(x)}}\delta{x}=how?$$ Please explain about this too.