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I'm a little bit confused about something that should actually be simple. If we have a function f between finite dimensional Banach spaces. Then we have the implications: If f is partially differentiable with continuous partial derivatives, then f is continuously differentiable, in particular, f is (totally) differentiable. However, the opposite implication is not true: There are functions that are differentiable, but don't have continuous partial derivatives.

My confusion is about how continuous differentiability ties in. Since the derivative of f in any point is given as a linear function, this function between (finite dimensional!) Banach spaces should be continuous. But the function that maps any point to it's derivative doesn't have to be linear, so it doesn't have to be continuous either. Is that right?

Otherwise (total) differentiability would imply continuous differentiability.

So my last question is: Is, then, continuous differentiability equivalent to partial continuous differentiability?

I feel silly even asking this, but I couldn't find any explicit explanation.

Amarus
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  • In infinite dimensions they are not equivalent, but in finite dimensions they are. A function is continuously totally differentiable if and only if its Jacobian matrix is continuous – Displayname Feb 08 '19 at 09:49
  • @Displayname It is true that for infinite dimensional spaces, it exists non continuous maps. However, a linear map that is the derivative at a point even in infinite dimension has to be continuous. This is by definition of the Fréchet derivative. – mathcounterexamples.net Feb 08 '19 at 12:55
  • When I say c.t.s differentiable I'm referring to the function $df: \mathbb{X} \rightarrow \mathcal{L}(\mathbb{X} ; \mathbb{Y})$ being a c.t.s function, this function isn't necessarily linear – Displayname Feb 08 '19 at 14:08

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Continuous differentiability is indeed equivalent to continuity of partial derivatives

See for example Proof that continuous partial derivatives implies differentiability for a proof of the reverse implication.

As a side comment, you should also notice that the Fréchet derivative supposes that the derivative (the linear map) is bounded. This is included in the definition even for infinite dimensional spaces.

And you’re right the continuity of the Fréchet derivative at a point doesn’t imply that the evolution around a point of the Fréchet derivative is continuous.

mathcounterexamples.net
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