Limits and Continuity in Two Variables A function of two variables, (x, y), is a function that can be expressed in the form f(x, y). The limit of a function...
Limits and Continuity in Two Variables
A function of two variables, (x, y), is a function that can be expressed in the form f(x, y). The limit of a function as (x, y) approaches a point (a, b) is the value that the function approaches as the point gets very close to (a, b).
In other words, the limit of f(x, y) as (x, y) approaches (a, b) is equal to the value of f(a, b).
Continuity
A function is continuous at a point (a, b) if the limit of the function as (x, y) approaches (a, b) is equal to the value of the function at (a, b).
In other words, a function is continuous if the graph of the function can be drawn without lifting the pen from the paper.
Examples
The function f(x, y) = x^2 + y^2 is continuous at the point (1, 1) because the limit of f(x, y) as (x, y) approaches (1, 1) is equal to the value of the function at (1, 1), which is 2.
The function f(x, y) = sin(x) is continuous at all points in its domain, which is the entire real line.
The function f(x, y) = 0 for all (x, y) is continuous at all points in its domain, which is the entire real line