# Limits and Continuity

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

#### 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

• lack of interruption or disconnection; the quality of being continuous in space or time

• a value to which a sequence or function converges

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

#### Figures

1. ##### Continuity

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

A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions (Figure 1). 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 $f(x,y) = \frac{x^2y}{x^4+y^2}$ approaches zero along any line through the origin. However, when the origin is approached along a parabola y = x2, it has a limit of 0.5. Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

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), continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f. As an example, consider

$f(x,y)= \begin{cases} \frac{y}{x}-y & \text{if } 1 \geq x > y \geq 0 \\ \frac{x}{y}-x & \text{if } 1 \geq y > x \geq 0 \\ 1-x & \text{if } x=y>0 \\ 0 & \text{else}. \end{cases}$.

It is easy to check that all real-valued functions (with one real-valued argument) that are given by fy(x):= f(x,y) are continuous in x (for any fixed y). Similarly, all fx are continuous as f is symmetric with regards to x and y. However, f itself is not continuous as can be seen by considering the squence f(1/n,1/n) (for natural n) which should converge to f(0,0)=0 if f was continuous. However, lim f(1/n,1/n) = 1.

#### Key Term Glossary

calculus
Any formal system in which symbolic expressions are manipulated according to fixed rules.
##### Appears in these related concepts:
continuity
lack of interruption or disconnection; the quality of being continuous in space or time
##### Appears in these related concepts:
converge
of a sequence, to have a limit
##### Appears in these related concepts:
domain
the set of all possible mathematical entities (points) where a given function is defined
##### Appears in these related concepts:
function
a relation in which each element of the domain is associated with exactly one element of the co-domain
##### Appears in these related concepts:
limit
a value to which a sequence or function converges
##### Appears in these related concepts:
multivariable
concerning more than one variable
##### Appears in these related concepts:
origin
the point at which the axes of a coordinate system intersect
##### Appears in these related concepts:
scalar
a quantity that has magnitude but not direction; compare vector
##### Appears in these related concepts:
scalar function
any function whose domain is a vector space and whose value is its scalar field
##### Appears in these related concepts:
variable
a quantity that may assume any one of a set of values