Limits and Continuity for Functions of Several Variables
From single-variable intuition to the richer landscape of multivariable analysis
The Definition Upgraded: From Intervals to Disks
Recall: The Single-Variable Definition
In single-variable calculus, we say $\lim_{x \to a} f(x) = L$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that:
$$ 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon $$The condition $0 < |x - a| < \delta$ describes a deleted interval — the set of points within distance $\delta$ of $a$, excluding $a$ itself. You can only approach $a$ from two directions: left and right.
The Upgrade: Two Variables
For a function $f(x,y)$, we say $\displaystyle\lim_{(x,y) \to (x_0, y_0)} f(x,y) = L$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that:
$$ 0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta \implies |f(x,y) - L| < \varepsilon $$🔑 The One Structural Change
The deleted interval $0 < |x - a| < \delta$ on the number line is replaced by a deleted open disk (deleted neighborhood) of radius $\delta$ centered at $(x_0, y_0)$ in the plane. Distance is now measured by the Euclidean distance $\sqrt{(x - x_0)^2 + (y - y_0)^2}$. Everything else — the logical structure, the role of $\varepsilon$ and $\delta$ — is identical.
⚠️ The Non-Negotiable Requirement
The limit $L$ must be a single number that works for all manners of approach to $(x_0, y_0)$. It must not depend on the direction, the path, or any other aspect of how $(x,y)$ reaches the point. This is the heart of the definition.
Visualizing the Deleted Neighborhood
Single Variable: Deleted Interval
Two Variables: Deleted Disk