Limits and Continuity for Multivariate Functions
Defining Limits of Two Variable functionsCase Studies in Two DimensionsContinuityThree or more VariablesLimits and Continuity for Multivariate FunctionsA. HavensDepartment of MathematicsUniversity of Massachusetts, AmherstFebruary 25, 2019A. HavensLimits and Continuity for Multivariate FunctionsDefining Limits of Two Variable functionsCase Studies in Two DimensionsContinuityThree or more VariablesOutline1Defining Limits of Two Variable functions2Case Studies in Two DimensionsAn Easy LimitFailure Along Different LinesLines Are Not EnoughAn Epsilon-Delta Game3ContinuityDefining ContinuitySome Continuous Functions4Three or more VariablesLimits and Continuity in Many VariablesDiscontinuities in Three DimensionsA. HavensLimits and Continuity for Multivariate FunctionsDefining Limits of Two Variable functionsCase Studies in Two DimensionsContinuityThree or more VariablesDefinition of a Limit in two VariablesDefinitionGiven a function of two variablesf:D R,D R2such thatDcontains points arbitrarily close to a point (a,b), we say that thelimit off(x,y) as (x,y) approaches (a,b) exists and has valueLifand only if for every real number >0 there exists a real number >0 such that|f(x,y) L|< whenever0< (x a)2+ (y b)2<.
Given a function of two variables f : D !R, D R2 such that D contains points arbitrarily close to a point (a;b), we say that the limit of f(x;y) as (x;y) approaches (a;b) exists and has value L if and only if for every real number ">0 there exists a real number >0 such that
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