Continuity

Suppose $f$ is a real-valued function defined on $D \subset \mathbb{R}^2$.

$f$ is continuous at $(a,b) \in D$ iff:

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{(x,y)\in D}\; \bigg(\sqrt{(x-a)^2 + (y-b)^2}<\delta\implies{|f(x,y)-f(a,b)|<\epsilon}\bigg)$$

• If $(a,b)\in D$ is an isolated point of $D$, then $f$ is continuous at $(a,b)$.
• Otherwise, for $f$ to be continuous at $(a,b) \in D$:
  • $f(a,b)$ must be well defined
  • $\lim_{(x,y)\to{(a,b)}} {f(x,y)}$ must exists
  • $\lim_{(x,y)\to{(a,b)}} {f(x,y)} = f(a,b)$

Note

• Constant multiples, sum, difference, product, quotient, and rational powers of continuous functions are continuous whenever they are well-defined.
• Polynomials in two variables are continuous functions.
• Rational functions are continuous functions provided they are well-defined.