I want to extend the concept of derivative of a real function of real variable to a function $f:A\subset \mathbb{R}^n \to \mathbb{R}^m$ with $A$ open. If $x_0 \in A$ then I say that $f$ has derivative $f'(x_0) \in \operatorname{Hom}(\mathbb{R}^n, \mathbb{R}^m)$ if
$$ \lim_{h\to 0} \frac{|f(x_0+h)-f(x_0)-f'(x_0)(h)|}{|h|}=0 .$$
When the function is given in explicit algebraic form as $f(x_1,\dots,x_n) = \sum \limits_{i=1}^m f_{i}(x_1,\dots,x_n)e_i$ and I know that the derivative exists at $x_0\in A$, then I can compute $f'(x_0)$ explicitly because
$$ [f'(x_0)]_{ij} = (D_jf_i)(x_0) $$
is the matrix representation of $f'(x_0)$ relative to standard bases and I know from basic calculus how to compute those partial derivatives.
My question is: if I can compute partial derivatives in $x_0$ without knowing if $f'(x_0)$ exists, is there some regularity condition on partial derivatives that is equivalent to the existence of $f'(x_0)$? The existence of partial derivatives isn't sufficient (for $n>1$), for example $\frac{xy}{x^2+y^2}$ has partial derivatives but isn't continuous in $(0,0)$ if is defined $0$ in $(0,0)$ and thus can't be differentiable there. Coming back to the general problem, if partial derivatives exist and are bounded in a neighborhood of $x_0$, then $f$ is continuous in $x_0$ but I believe it could be not differentiable, although I can't write a counterexample. If the partial derivatives are continuous in $x_0$ then $f'(x)$ should exist in a neighborhood of $x_0$ and should be continuous in $x_0$, but this is clearly more than differentiability in $x_0$.
NEWS Differentiability seems to be a slippery regularity. I discovered from Rudin "Principles of Mathematical Analysis" that an equivalence exists for continuously differentiable functions, in fact $f\in C^1(A)$ if and only if $D_jf_i\in C^1(A)$ for all $i,j$. The question seems difficult because of the fact that the regularity of a multivariable function and that of its partial derivatives seem to have a weak connection, although a general heuristic principle for this type of problems could be: a stronger regularity on partial derivatives implies a weaker regularity on the function.
Best Answer
I guess you are looking for the below theorem: Reference Mathematical Analysis, by Tom Apostol page $357$.