Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let "f" : "A" → "F" be a continuous function from an open set "A" in a Banach space "E" to another Banach space "F". Then the quasi-derivative of "f" at "x" ∈ "A" is a linear transformation "u" : "E" → "F" with the following property: for every continuous function "g" : [0,1] → "A" with "g"(0)="x" such that "g"′(0) ∈  E exists, If such a linear map "u" exists, then "f" is said to be "quasi-differentiable" at "x". Continuity of "u" need not be assumed, but it follows instead from the definition of the quasi-derivative. If "f" is Fréchet differentiable at "x", then by the chain rule, "f" is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at "x". The converse is true provided "E" is finite-dimensional. Finally, if "f" is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.