I saw this in a book explaining the Infeasible Newton Method. It used this to prove an inequality which is then used to prove the convergence of the method. Can anyone explain how it is derived and perhaps explain some intuition behind it?
My intuition so far is that we can write $f(x+h)$ as $f(x+\sum_i^N{rh})$ where $r=\frac{1}{N}$. This would then imply the approximation $$f(x+h)=f(x+\sum_i^N{rh})\approx f(x+\sum_2^N{rh})+Df(x+\sum_2^N{rh})rh\approx f(x)+Df(x)rh+...+Df(x+\sum_2^N{rh})rh$$ Now since we let $N\to\infty$ and $r\to0$ the sum basically becomes an integral from 0 to 1. Is this right and what is the formal derivation?