Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
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 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
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 converge
 Appears in these related concepts: Comparison Tests, Summing an Infinite Series, and Convergence of Series with Positive Terms
 domain
 Appears in these related concepts: Double Integrals Over General Regions, The Derivative as a Function, and Finding the Domain of a Rational Function
 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, Applications of Minima and Maxima in Functions of Two Variables, and Double Integrals Over Rectangles
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Types of Muscle Tissue, and ThreeDimensional Coordinate Systems
 scalar
 Appears in these related concepts: VectorValued Functions, Calculus of VectorValued Functions, and Multiplying Vectors by a Scalar
 sequence
 Appears in these related concepts: Series, Introduction to Sequences, and Finding the General Term
 variable
 Appears in these related concepts: Related Rates, Controlling for a Variable, and Math Review
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Limits and Continuity.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 30 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/partialderivatives21/limitsandcontinuity1472863/