I learn that the sufficient condition for Lipschitz smoothness is the function $f$ satisfys that: $$ \mid f(x+h)-f(x)-\langle\xi,h\rangle \mid \leq c\parallel h \parallel ^2 $$ where $\xi = f'$
I want to use this inequality to bound the difference between $f(x+h)$ and $f(x)+\langle\xi,h\rangle$, then what condition should be satisfied for the function $f$ in order to apply the inequality? I see someone simply assume that $f$ is sufficiently smooth before applying the inequality, is it correct?
What also confuse me is that the description above seems to have nothing to do with Lipschitz smoothness because the inequality above is the sufficient condition for Lipschitz smoothness rather than a property, basically I can't say the inequality holds because the function $f$ is Lipschitz smooth. As a result, the question is what's the property of Lipschitz smoothness, what's the application and how to use it?