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hivesaeed4
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Where is the function
$${f (x, y) = x^2 + x y + y^2}$$
continuous?
How do I go about solving such problems?
$${f (x, y) = x^2 + x y + y^2}$$
continuous?
How do I go about solving such problems?
Continuity in multi-variable calculus refers to the property of a function where small changes in the input variable result in small changes in the output variable. In other words, if the inputs to a function are close together, the corresponding outputs should also be close together.
In multi-variable calculus, a function is considered continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point. This means that as the input values approach the given point, the output values also approach the value of the function at that point.
Continuity is important in multi-variable calculus because it allows us to study the behavior of functions at specific points and make predictions about their behavior in the surrounding area. It also allows us to use tools such as derivatives and integrals to analyze and solve problems involving multi-variable functions.
Yes, a function can be continuous at a specific point but not in its entire domain. This means that the function may have a break or discontinuity at one or more points, but it is still considered continuous at the specific point in question.
To determine if a function is continuous in multi-variable calculus, we can use the definition of continuity to check if the limit of the function exists and is equal to the value of the function at the given point. We can also use graphical and algebraic methods, such as the intermediate value theorem and the epsilon-delta definition, to verify continuity.