Originally Posted by

**Jose27** It's easy to see that $\displaystyle c*h = <x,h>$ ie. $\displaystyle <x,h>=D_f(x)h$ use this and the fact that $\displaystyle \Vert x+h \Vert ^2 = ^\Vert x \Vert ^2 + 2<x,h> + \Vert h \Vert ^2$ and substitute in the definition: $\displaystyle \vert \frac{ f(x+h) - f(x) - D_f(x)h}{ \Vert h \Vert } \vert$ and see that this goes to zero as $\displaystyle \Vert h \Vert \rightarrow 0$. ($\displaystyle <,>$ denotes the interior product, I'm assuming you're using the usual norm in $\displaystyle \mathbb{R} ^n$)