Limits and Continuity of Functions of Two Variables

Summary

For $f(x,y)$ , the limit as $(x,y)\to (a,b)$ must approach the same value along every path in the plane. Different path limits prove nonexistence; matching on several paths never proves existence by itself. Continuity requires the limit to exist and equal the function value.

Prerequisites

Limits, Multivariable Functions

Definition

means: for every $\epsilon >0$ there exists $\delta >0$ such that

The function $f$ is continuous at $(a,b)$ if $f(a,b)$ is defined and

Conditions / Assumptions

• Direct substitution is valid when $f$ is known continuous at $(a,b)$ (polynomials, rational functions off their zero-denominator sets, compositions of continuous functions).
• Path tests: if two paths give different limits, the two-variable limit does not exist. Agreement on many paths is inconclusive for existence.

Worked Example

Consider

• Along $y=0$ : $\underset{x\to 0}{lim}f(x,0)=1$ .
• Along $y=x$ : $\underset{x\to 0}{lim}f(x,x)=0$ .

The path limits disagree, so $\underset{(x,y)\to (0,0)}{lim}f(x,y)$ does not exist. Defining $f(0,0)=1$ (or any value) cannot make $f$ continuous at the origin.

By contrast, $g(x,y)=x^2+y^2$ is continuous everywhere by the polynomial continuity theorem, so $\underset{(x,y)\to (a,b)}{lim}g(x,y)=a^2+b^2$ by substitution.

Common Mistakes

• Using direct substitution when the expression is undefined or the candidate is not continuous.
• Concluding that a limit exists because it is the same along lines $y=mx$ only.
• Typo-level confusion between continuity ( $L=f(a,b)$ ) and mere existence of $L$ .

Connections

• Next: Partial Derivatives, Differentiability of a Function
• Path issues reappear for directional derivatives vs total differentiability

References

Multivariable limits and continuity are treated in OpenStax Calculus Volume 3.^[1]