Question

State the definition of continuity of a function of two variables.

Short answer

A function f(x, y) of two variables is continuous at a point (a, b) if for every positive number ε, there exists a positive number δ such that if the distance between (x, y) and (a, b) is less than δ, then |f(x, y) - f(a, b)| is less than ε. That is, the limit of f(x, y) as (x, y) approach (a, b) is equal to the function's value at (a, b).

Step-by-step solution

Define a Function of Two Variables

A function of two variables, \(f(x, y)\), takes two input values and produces a single output value. The input values are typically represented by the variables \(x\) and \(y\). Hence, the function operates on ordered pairs of real numbers \((x, y)\) to produce real numbers as output.

Describe the Concept of Continuity

Continuity is a fundamental concept in calculus. It describes a function that does not have any jumps, breaks, or holes. In one-variable calculus, a function \(f(x)\) is continuous at a point \(x=a\) if the limit of the function as \(x\) approaches \(a\) is equal to the function's value at \(a\). This concept extends to functions of two variables.

Define Continuity for a Function of Two Variables

A function \(f(x, y)\) of two variables is continuous at a point \((a, b)\) if for every positive number \(\varepsilon\), there exists a positive number \(\delta\) such that if the distance between \((x, y)\) and \((a, b)\) is less than \(\delta\), then the absolute value of f(x, y) - f(a, b) is less than \(\varepsilon\). That is, \[ \lim_{{(x, y) \to (a, b)}} f(x, y) = f(a, b) \]