Fréchet derivative

context	$X,Y$ … Banach spaces with topology
context	$\mathcal O$ … open in $X$
definiendum	$D:\mathrm{Continuous}(\mathcal O,Y)\to \mathrm{Continuous}(\mathcal O,\mathrm{BoundedLinOp}(X,Y))$
definiendum	$Df:=x\mapsto J_x^f$

This definition does nothing more than emphasizing the functionality of $L_x^f$ in $f$ .