Recall of Limit and Continuity of Functions of One Variable

Before extending limits and continuity to two variables, it helps to restate the one-variable definitions (from Class XI) in the language of neighbourhoods, since that is the form that generalizes cleanly.

Recall — Definition (Limit, one variable). $f:(a,b)\to\mathbb R$ has limit $L$ at $x_0\in(a,b)$, written $\displaystyle\lim_{x\to x_0}f(x)=L$, if: for every neighbourhood $(L-\varepsilon,L+\varepsilon)$, $\varepsilon>0$, of $L$, there exists a neighbourhood $(x_0-\delta,x_0+\delta)\subset(a,b)$, $\delta>0$, of $x_0$, such that $f(x)\in(L-\varepsilon,L+\varepsilon)$ whenever $x\in(x_0-\delta,x_0+\delta)\setminus\{x_0\}$. Equivalently, in modulus notation: for every $\varepsilon>0$ there is $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-x_0|<\delta$.

This is also characterised by matching one-sided limits: $f$ has a limit $L$ at $x_0$ (where $f$ is defined near, but not necessarily at, $x_0$) iff the right-hand limit $L_1=\lim_{x\to x_0^+}f(x)$ exists, the left-hand limit $L_2=\lim_{x\to x_0^-}f(x)$ exists, and $L_1=L_2$ (then $L=L_1=L_2$). Continuity at $x_0$ (where $f(x_0)$ is defined) additionally requires $L=f(x_0)$.

Neighbourhoods in the plane. The one-variable neighbourhood of $x_0\in\mathbb R$ is the interval $(x_0-r,x_0+r)$, $r>0$. To do the same for a point $(u,v)\in\mathbb R^2$, define the $r$-neighbourhood of $(u,v)$ as the open disc

$B_r((u,v)) = \big\{(x,y)\in\mathbb R^2 \ \big|\ (x-u)^2+(y-v)^2 < r^2\big\},$ …