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Chapter 13: Problem 7

What three conditions must be met for a function \(f\) to be continuous at the point \((a, b) ?\)

Short Answer

Expert verified
Answer: For a function, f, to be continuous at a point (a, b), it must meet the following three conditions: 1) The function is defined at the point, 2) The limit of the function exists at the point, and 3) The function's value equals its limit at the point.

Step by step solution

01

Definition of Continuity

A function \(f\) is defined to be continuous at a point \((a, b)\) if it satisfies three conditions:
02

Condition 1: The function is defined at the point

The function \(f\) must be defined at the point \((a, b)\) which means the order pair \((a, b)\) is in the domain of the function \(f\).
03

Condition 2: The limit of the function exists at the point

The limit of the function \(f\) as it approaches the point \((a, b)\) must exist. This can be expressed as follows: \[\lim_{x \to a, y \to b} f(x, y)\]
04

Condition 3: The function's value equals its limit at the point

The function's value at the point \((a, b)\) must be equal to the limit of the function as it approaches \((a, b)\). This can be expressed as: \[f(a, b) = \lim_{x \to a, y \to b} f(x, y)\] If these three conditions are met, then the function \(f\) is continuous at the point \((a, b)\).

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