here's my problem: attending Calculus II class I ran into this proof that is not clear to me (our teacher takes so many things as obvious when they're actually not).
Theorem: Let $f:D\subset\mathbb{R}^n \to \mathbb{R}$ be a function such that it admits all partial derivatives and they're continuous around a point $x_0 \in D$ then $f$ is differentiable in $x_0$.
Proof: ( My attempt, cause at half of it, my teacher chokes it saying that a thing it implies differentiability ):
We're showing this in the specific case $n=2$, where $\vec{x_0} = (x,y)$ and $\vec{h} = (h_x, h_y)$.
The following identity is valid (we add and subtract $f(x,y+h_y)$ ):
$f(x+h_x, y+h_y) - f(x,y) = f(x+h_x, y+h_y) - f(x,y+h_y) + f(x,y+h_y) - f(x,y)$
We can recognize here that
$f(x+h_x, y+h_y) - f(x,y+h_y) = h_x\frac{\partial f}{\partial x}(x,y+h_x) + o(h_x)$
and similarly
$f(x,y+h_y) - f(x,y) = h_y\frac{\partial f}{\partial y}(x,y) + o(h_y)$
It follows that
$f(x+h_x, y+h_y) - f(x,y) = h_x\frac{\partial f}{\partial x}(x,y+h_x) + o(h_x) + h_y\frac{\partial f}{\partial y}(x,y) + o(h_y)$
Adding and subtracting from the RHS $h_x\frac{\partial f}{\partial x}(x,y)$ we get
$f(x+h_x, y+h_y) - f(x,y) = h_x\frac{\partial f}{\partial x}(x,y+h_x) -h_x\frac{\partial f}{\partial x}(x,y) + h_x\frac{\partial f}{\partial x}(x,y) + h_y\frac{\partial f}{\partial y}(x,y) + o(h_x) + o(h_y)$
Which is equivalent to this:
$ f(x+h_x, y+h_y) - f(x,y) = h_x[\frac{\partial f}{\partial x}(x,y+h_x) -\frac{\partial f}{\partial x}(x,y)] + \nabla f|_{(x,y)}\cdot h + o(h_x) + o(h_y)$
By definiton of differential:
$ f(x+h_x, y+h_y) - f(x,y) = h_x[\frac{\partial f}{\partial x}(x,y+h_x) -\frac{\partial f}{\partial x}(x,y)] + df(x,y) + o(h_x) + o(h_y)$
Here $\frac{\partial f}{\partial x}(x,y+h_x) -\frac{\partial f}{\partial x}(x,y)$ goes to $0$ for small enough $h_x$ and then, being the partial derivative continuous, this whole thing goes to $0$ (I'm not really sure about this, can I say it even without passing through the limit?).
Then from the previous we have that $ f(x+h_x, y+h_y) - f(x,y) = df(x,y) + o(h_x) + o(h_y)$ which allows us to assert that the hypothesis is true.
This too is not clear at all to me: Somehow $o(h_x) + o(h_y) \to o(\begin{Vmatrix}\vec{h}\end{Vmatrix})$? Why?