I have trouble understanding the general concept of how to prove the continuity of the partial derivative (I will here use the continuity definition via limits of sequences). May be someone can give me a quick feedback on my thoughts:
If I take the partial derivative of a function $f$ two things might happen:
1.) The function is simply a compostion of differentiable functions and not piece-wise defined, say $f:\mathbb{R}^2 \to \mathbb{R}$, $f{x \choose y}=x^2+xy^2$. So applying the rules of differentiation yields the partial derivative w.r.t $x$: $D_xf=2x+y^2$. If I check for continuity I plug in an arbitrary point ${x \choose y}$ into $D_xf{x \choose y}$ and afterwards I check if for any convergent sequence $lim_{n \to \infty}{x_n \choose y_n}= {x \choose y}$ it holds $D_xf{x \choose y}=lim_{n \to \infty}D_xf{x_n \choose y_n}$. Or I simply argue that $2x+y^2$ is a composition of continuous functions which also is continuous. This, I would regard as the "standard" case.
2.) However, I get confused when the function $f$ is piece-wise defined, as this almost always means that there are boundary points, say ${1 \choose 1}$, were you cannot simply apply the rules of differentiation but must check the limit. Let's assume a continuous function $g:\mathbb{R}^2 \to \mathbb{R}$,
$$g{x \choose y}=\left\{\begin{array}{ll} x^2+xy^2, & {x \choose y} \neq {1 \choose 1} \\ 2, & {x \choose y} = {1 \choose 1}\end{array},\right. $$ Let the partial w.r.t. $x$ at ${1 \choose 1}$ be: $D_xg{1 \choose 1}= lim_{x\to 1} \frac{x^2+x1^2-2}{x-1}= 3$. Applying the rules of differentiation yields the partial for all other points except ${1 \choose 1}$ is: $D_xg{x \choose y}=2x+y^2$.
If I now want to check for continuity of the partial at point ${1 \choose 1}$ then for any convergent sequence $lim_{n \to \infty}{x_n \choose y_n}= {1 \choose 1}$ which I plug in into $D_xg{x \choose y}=2x+y^2$ it must hold $D_xg{1 \choose 1}=3=lim_{n \to \infty}D_xg{x_n \choose y_n}= lim_{n \to \infty} (2x_n+y_n^2)$.
I hope this is not too wordy but I would appreciate if someone could tell me if my thoughts on that topic are correct.