I am confused about the definition of continuity and differentiability of n-variable functions. Also, what is the meaning of continuously differentiable?

Here are some examples that I don't understand fully. Please explain.

1. Let function f be defined on the whole xy plane. f(x,y)= 1 if x=y=/=0 and f(x,y)=0 otherwise. In this case, f is not continuous at (0,0) but both partial derivatives fx and fy exists at (0,0).

2. Let function f be defined as f(x,y)=(x^1/3 + y^1/3)^3. f(x,y) is continuous and has partial derivatives at origin (0,0) but is not differentiable there.

3. Let f be defined by f(x,y)=y^2 + x^3 sin (1/x) for x=/=0, and f(0,y)=y^2. f is differentiable at (0,0), but is not continuously differentiable there because fx(x,y) is not continuous at (0,0).

Thanks. > <