Calculus
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Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 7
Created by Boundless

Limits and Continuity

A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.

Learning Objective

  • Describe the relationship between the multivariate continuity and the continuity in each argument


Key Points

    • The function f(x,y)=x2yx4+y2f(x,y) = \frac{x^2y}{x^4+y^2}f(x,y)=​x​4​​+y​2​​​​x​2​​y​​ 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 counter-intuitive results not demonstrated by single-variable 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 f(x,y)=x2yx4+y2f(x,y) = \frac{x^2y}{x^4+y^2}f(x,y)=​x​4​​+y​2​​​​x​2​​y​​ approaches zero along any line through the origin. However, when the origin is approached along a parabola y=x2y = x^2y=x​2​​, it has a limit of 0.50.50.5. Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

Continuity

Continuity in single-variable function as shown is rather obvious. However, continuity in multivariable functions yields many counter-intuitive results.

Continuity in each argument does not imply multivariate continuity. For instance, in the case of a real-valued function with two real-valued parameters, f(x,y)f(x,y)f(x,y), continuity of fff in xxx for fixed yyy and continuity of fff in yyy for fixed xxx does not imply continuity of fff. As an example, consider

f(x,y)={yx−yif 1≥x>y≥0xy−xif 1≥y>x≥01−xif x=y>00else.f(x,y)= \begin{cases} \displaystyle{\frac{y}{x}}-y & \text{if } 1 \geq x > y \geq 0 \\ \displaystyle{\frac{x}{y}}-x & \text{if } 1 \geq y > x \geq 0 \\ 1-x & \text{if } x=y>0 \\ 0 & \text{else}. \end{cases}f(x,y)=​⎩​⎪​⎪​⎪​⎪​⎪​⎨​⎪​⎪​⎪​⎪​⎪​⎧​​​​x​​y​​−y​​y​​x​​−x​1−x​0​​​if 1≥x>y≥0​if 1≥y>x≥0​if x=y>0​else.​​

It is easy to check that all real-valued functions (with one real-valued argument) that are given by fy(x)=f(x,y)f_y(x)= f(x,y)f​y​​(x)=f(x,y) are continuous in xxx (for any fixed yyy). Similarly, all fxf_xf​x​​ are continuous as fff is symmetric with regards to xxx and yyy. However, fff itself is not continuous as can be seen by considering the sequence f(1n,1n)f \left(\frac{1}{n},\frac{1}{n} \right)f(​n​​1​​,​n​​1​​) (for natural nnn), which should converge to f(0,0)=0\displaystyle{f (0,0) = 0}f(0,0)=0 if fff is continuous. However, limf(1n,1n)=1\lim f \left(\frac{1}{n},\frac{1}{n} \right) = 1limf(​n​​1​​,​n​​1​​)=1.

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