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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

scalar function
any function whose domain is a vector space and whose value is its scalar field

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
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 singlevariable 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,
It is easy to check that all realvalued functions (with one realvalued argument) that are given by
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Key Term Reference
 converge
 Appears in these related concepts: Comparison Tests, Convergence of Series with Positive Terms, and Infinite Geometric Series
 domain
 Appears in these related concepts: Introduction to Rational Functions, Restricting Domains to Find Inverses, and Composition of Functions and Decomposing a Function
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 multivariable
 Appears in these related concepts: Functions of Several Variables, Applications of Minima and Maxima in Functions of Two Variables, and Double Integrals Over Rectangles
 origin
 Appears in these related concepts: Types of Muscle Tissue, Lever Systems, and ThreeDimensional Coordinate Systems
 scalar
 Appears in these related concepts: VectorValued Functions, Superposition of Electric Potential, and Addition and Subtraction; Scalar Multiplication
 sequence
 Appears in these related concepts: Summing an Infinite Series, Introduction to Sequences, and The General Term of a Sequence
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Limits and Continuity.” Boundless Calculus. Boundless, 08 Aug. 2016. Retrieved 29 Aug. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/partialderivatives21/limitsandcontinuity1472863/