id: 06575085
dt: j
an: 2016c.00861
au: Baderian, Armen; Javadi, Mohammad
ti: Proof of the continuity at a point for a class of functions of two
independent variables by construction.
so: Math. Comput. Educ. 50, No. 1, 52-56 (2016).
py: 2016
pu: MATYC Journal, Old Bethpage, NY
la: EN
cc: I65 I25 E55
ut: multivariable calculus; polynomial functions; continuity at a point;
proofs; circles; limits; heuristics; concept formation; approach
ci:
li:
ab: From the text: When a new mathematical topic is to be presented in the
classroom, examples that demonstrate the significant related concepts
should be included. However, examples for some topics are deficient,
lengthy, or nonexistent. Consider a lesson for a multivariable calculus
class on the continuity of a function of two independent variables at a
point of its domain. A unique limit of the function at a point of the
domain is necessary. Examples presented in many texts primarily
demonstrate the process of showing when a limit does not exist. For
these types of examples, one simply shows the existence of two
different limits of a function that are obtained along two different
paths to the given point in the domain, respectively. A proof
establishing the existence of a unique limit along all possible paths
to that given domain point is sometimes briefly described. Thus some
students are probably not directly experiencing the derivation of the
existence of a unique limit. This article presents a direct proof that
establishes the continuity of any multi-variable polynomial function at
any given domain point. It is a method that constructs a mathematical
statement that is used to prove the existence of a unique limit.
Students can use this method to prove continuity at a point, and thus
experience, in a positive sense, the manipulation of the mathematical
structure of an epsilon-delta argument. This article may be of interest
to instructors and students of multivariable calculus.
rv: