Understanding Continuity in Multivariable Functions: Key Definitions and Theorems

Continuity of Multivariable Functions

Definition 1. Let a multivariable function \(f\) be defined on a domain \(D\) and let \(a\) be a cluster point of \(D\). If the limit \(\lim_{x\to a} f(x)\) exists, then \(f\) is continuous at \(a\); otherwise \(a\) is called a point of discontinuity.

If a function is continuous at every point of \(D\), it is said to be continuous on \(D\).

Example: The function \(f(x,y)=\frac{x^2-y^2}{x^2+y^2}\) has no limit at the origin, so the origin is a discontinuity point. Another example: the function \(g(x,y)=\frac{1}{x^2+y^2}\) is discontinuous on the circle \(x^2+y^2=0\).

Conclusion: All elementary multivariable functions are continuous on their domains of definition.

For multivariable continuous functions on a closed bounded domain, the following properties hold (analogous to one‑variable results):

(1) Boundedness theorem: the function is bounded on the domain.

(2) Extreme‑value theorem: the function attains a maximum and a minimum on the domain.

(3) Intermediate‑value theorem: for any value between the function’s minimum and maximum, there exists a point in the domain where the function takes that value.

(4) Uniform‑continuity theorem: the function is uniformly continuous on the domain.

Example 1: Compute the limit \(\lim_{(x,y)\to(0,0)} \frac{x^2-y^2}{x^2+y^2}\).

Solution: (Detailed computation omitted for brevity.)

Example 2: Determine the continuity domain of the function \(h(x,y)=\frac{x}{x^2+y^2}\).

Solution: (Detailed analysis omitted for brevity.)