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Definition: \(n\) times Continuously Differentiable Functions

Let \((V,\| \|)\) and \((W,\| \|)\) be finitely dimensional normed vector spaces, \(G\subseteq V\) an open subset, and \(f\colon G\rightarrow W\) a function. Furthermore, let \(x\in G\) be a point and \(v_{1},\ldots ,v_{n}\) be fixed vectors in \(V\). Then we say the function \(f\) is \(n\) times continuously differentiable, if

for any permutation \(\pi(i)\) of the vectors \(v_i\), \(i=1,2,\ldots,n\), the higher order directional derivative \(D_{v_{\pi(n)}}(...D_{v_{\pi(2)}}(D_{v_{\pi(1)}}f(x)))\) exists and
this higher order directional derivative is continuous.

Corollaries: 1
Definitions: 2 3
Proofs: 4 5 6 7
Propositions: 8 9
Theorems: 10

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References

Adapted from CC BY-SA 3.0 Sources:

Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück