多元函数中的连续性

Continuity means the limit lands on the value

Removable breaks versus permanent breaks

Algebra and composition assemble continuity

Continuity is the license for substitution

The function $f(x, y) = \dfrac{x^2 - y^2}{x^2 + y^2}$ has no value at the origin. Can $f(0,0)$ be defined so that $f$ becomes continuous there?

Why is $f(x, y) = \sin(x^2 + y^2)$ continuous at every point of the plane?

At which points is $f(x, y) = \dfrac{xy}{x - y}$ continuous?

The function $g(x, y) = \dfrac{x^3 - y^3}{x^2 + y^2}$ has a removable break at the origin. What value should be assigned to $g(0,0)$ to make $g$ continuous there, as a plain number?

Compute $\lim_{(x,y) \to (1,2)} \dfrac{xy + x^2}{x + y}$ and give the result as a plain number or fraction like 3/2.