Differentiability in Higher Dimensions
See also: Multivariable calculusA function f: Rm → Rn is said to be differentiable at a point x0 if there exists a linear map J: Rm → Rn such that
If a function is differentiable at x0, then all of the partial derivatives must exist at x0, in which case the linear map J is given by the Jacobian matrix. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus.
Note that existence of the partial derivatives (or even all of the directional derivatives) does not guarantee that a function is differentiable at a point. For example, the function ƒ: R2 → R defined by
is not differentiable at (0, 0), but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function
is not differentiable at (0, 0), but again all of the partial derivatives and directional derivatives exist.
It is known that if the partial derivatives of a function all exist and are continuous in a neighborhood of a point, then the function must be differentiable at that point, and is in fact of class C1.
Read more about this topic: Local Linearity
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