Summary of continuity

Basic. Polynomials and quotients of continuous functions (where denominator nonzero) are continuous.

Continuity at a point. Show $f(x,y)\to f(x_0,y_0)$; use bounding, polar coordinates, or the squeeze theorem.

Discontinuity. Exhibit two paths with different limits, or a path where the limit diverges or doesn't exist.

Notable counterexamples. $f(x,y)=\frac{x}{x+y}$ at origin, $f(x,y) = \sin(1/(x^2+y^2))$ at origin.

Key theorems. Continuous image of compact is compact; continuous on compact is uniformly continuous (Heine–Cantor); norm and projections are continuous.