Both in my textbook and on Wikipedia, continuous differentiability of a function $f:\Bbb R^m \to \Bbb R^n$ is defined by the existence and continuity of all of the partial derivatives. Since there is a notion of a (total) derivative (AKA differential) for multivariable functions, I'm wondering why continuous differentiability is not defined as existence and continuity of the derivative map $Df(a)$? Is there some reason why having existence and continuity of partials is more convenient or maybe continuity of the total derivative is too strict of a condition?
Answer
Continuous differentiability of the function $f: \mathbb{R}^m \to \mathbb{R}^n$ (in terms of partial derivatives) is equivalent to existence and continuity of the map
$$Df: \mathbb{R}^m \to L(\mathbb{R}^m, \mathbb{R}^n)$$
$$ x \to Df_x$$
which takes a point to the derivative at the point. Any book on analysis on $\mathbb{R}^n$ will have a proof of this fact.
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