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Differentiable Function

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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