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Limits and Continuity
A study of limits and continuity in multivariable calculus yields counterintuitive results not demonstrated by singlevariable functions.
Learning Objective

Describe the relationship between the multivariate continuity and the continuity in each argument
Key Points
 The function
$f(x,y) = \frac{x^2y}{x^4+y^2}$ has different limit values at the origin, depending on the path taken for the evaluation.  Continuity in each argument does not imply multivariate continuity.
 When taking different paths toward the same point yields different values for the limit, the limit does not exist.
Terms

continuity
lack of interruption or disconnection; the quality of being continuous in space or time

limit
a value to which a sequence or function converges

scalar function
any function whose domain is a vector space and whose value is its scalar field
Full Text
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by singlevariable functions . For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola. For example, the function
Continuity
Continuity in single variable function as shown is rather obvious. However, continuity in multivariable functions yields many counterintuitive results.
Continuity in each argument does not imply multivariate continuity: For instance, in the case of a realvalued function with two realvalued parameters, f(x,y), continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f. As an example, consider
It is easy to check that all realvalued functions (with one realvalued argument) that are given by f_{y}(x):= f(x,y) are continuous in x (for any fixed y). Similarly, all f_{x} are continuous as f is symmetric with regards to x and y. However, f itself is not continuous as can be seen by considering the squence f(1/n,1/n) (for natural n) which should converge to f(0,0)=0 if f was continuous. However, lim f(1/n,1/n) = 1.
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Key Term Reference
 converge
 Appears in these related concepts: The Natural Logarithmic Function: Differentiation and Integration, Indeterminate Forms and L'Hôpital's Rule, and Convergence of Series with Positive Terms
 domain
 Appears in these related concepts: The Derivative as a Function, Composition of Functions and Decomposing a Function, and Phenotypic Analysis
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 multivariable
 Appears in these related concepts: Functions of Several Variables, Tangent Planes and Linear Approximations, and Applications of Minima and Maxima in Functions of Two Variables
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Overview of Different Muscle Functions, and ThreeDimensional Coordinate Systems
 scalar
 Appears in these related concepts: VectorValued Functions, Multiplying Vectors by a Scalar, and Superposition of Electric Potential
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Limits and Continuity.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/partialderivatives21/limitsandcontinuity1472863/