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 $f$ $f$ to be continuous at $(a,b) \in D$ $(a, b) \in D$ :
- $f(a,b)$ must be well defined
- $\lim_{(x,y)\to{(a,b)}} {f(x,y)}$ must exists
- $\lim_{(x,y)\to{(a,b)}} {f(x,y)} = f(a,b)$

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September 29, 2026 1 min read

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