Differentiable Function

Posted November 22nd, 2007 by Isoscel

Let us consider $(X,||\: ||_X)$ and $(Y,|| \:||_Y)$ two normed vector spaces , $D$ an open set in $X$ $f:D\subseteq X\rightarrow Y$ a function and $a\in D$ . We shall say that f is differentiable in a if and only if there is linear continuous operator $T\in \mathcal{L}_c(X,Y)$ such as

$\lim_{x\rightarrow a}\frac{f(x)-f(a)-T(x-a)}{||x-a||_X}=0$

If such a linear continuous operator exists , it is unique and we call $T=df(a)$ the differential of f in a.
If f is differentiable in any $x\in D$ we say f is differentiable on $D$ and so there is a function

$df:D\subseteq X\rightarrow \mathcal{L}_c(X,Y)$

called the first differential of f.

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