2.3 Limits and continuity in multiple variables

Limits of Multivariable Functions

In single-variable calculus, a limit only requires approaching from two directions: left and right. For functions of several variables, $(x,y)$ can approach a point from infinitely many directions and along curved paths. This makes proving a limit exists much harder, and it opens up new ways for limits to fail.

Evaluating Limits and Directional Limits

The limit of a multivariable function $f(x,y)$ as $(x,y)$ approaches a point $(a,b)$ is written as:

$\lim_{(x,y)\to(a,b)}f(x,y)=L$

For this limit to exist, $f(x,y)$ must approach the same value $L$ regardless of the path taken toward $(a,b)$. That's the key difference from single-variable limits.

A directional limit is the limit of $f$ as $(x,y)$ approaches $(a,b)$ along one specific path. If two different paths give different values, the overall limit does not exist.

Example: Consider $\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+y^2}$.

• Along the path $y = mx$ (straight lines through the origin), substitute to get $\frac{mx^2}{x^2+m^2x^2} = \frac{m}{1+m^2}$, which depends on $m$. Different slopes give different values.
• Along $y = x$, the limit is $\frac{1}{2}$. Along $y = 0$, the limit is $0$.
• Since two paths yield different results, the limit does not exist.

A path limit generalizes this idea further: you parametrize the approach using a curve. For instance, $\lim_{t\to0}f(t\cos t,\, t\sin t)$ evaluates the limit along the spiral $x=t\cos t,\, y=t\sin t$ as $t\to0$. Parametric paths let you test more exotic approaches beyond straight lines.

Formal Definition and Iterated Limits

The $\varepsilon$-$\delta$ definition for multivariable limits:

$\lim_{(x,y)\to(a,b)}f(x,y)=L$ means that for every $\varepsilon>0$, there exists a $\delta>0$ such that

$|f(x,y)-L|<\varepsilon \quad \text{whenever} \quad 0<\sqrt{(x-a)^2+(y-b)^2}<\delta$

The distance $\sqrt{(x-a)^2+(y-b)^2}$ is the Euclidean distance from $(x,y)$ to $(a,b)$, so this captures all directions at once. The condition $0 < \sqrt{(\cdots)}$ excludes the point $(a,b)$ itself, just like in single-variable limits.

Iterated limits evaluate one variable at a time while holding the other fixed:

$\lim_{x\to a}\left(\lim_{y\to b}f(x,y)\right) \quad \text{and} \quad \lim_{y\to b}\left(\lim_{x\to a}f(x,y)\right)$

Be careful with the relationship between iterated limits and the full multivariable limit:

• If the multivariable limit $L$ exists and both iterated limits exist, then both iterated limits equal $L$.
• If the two iterated limits exist but are not equal, the multivariable limit does not exist.
• If the two iterated limits are equal, the multivariable limit still might not exist. Agreement of iterated limits is necessary but not sufficient.

Example: For $f(x,y)=\frac{xy}{x^2+y^2}$ near $(0,0)$:

• $\lim_{x\to 0}\left(\lim_{y\to 0}\frac{xy}{x^2+y^2}\right) = \lim_{x\to 0}\frac{0}{x^2} = 0$
• $\lim_{y\to 0}\left(\lim_{x\to 0}\frac{xy}{x^2+y^2}\right) = \lim_{y\to 0}\frac{0}{y^2} = 0$

Both iterated limits equal $0$, yet the multivariable limit does not exist (as shown by the path test above). This is a classic example of why iterated limits alone can't confirm a multivariable limit.

Continuity in Multiple Variables

Continuity and Partial Continuity

A function $f(x,y)$ is continuous at $(a,b)$ if three conditions hold:

1. $f(a,b)$ is defined.
2. $\lim_{(x,y)\to(a,b)}f(x,y)$ exists.
3. $\lim_{(x,y)\to(a,b)}f(x,y) = f(a,b)$.

Polynomials in $x$ and $y$, rational functions (where the denominator is nonzero), and compositions of continuous functions are all continuous on their domains. For example, $f(x,y)=x^2+y^2$ is continuous everywhere.

Partial continuity (also called separate continuity) is a weaker condition. A function is partially continuous in $x$ at $(a,b)$ if $\lim_{x\to a}f(x,b)=f(a,b)$, treating $y$ as the constant $b$. Similarly for $y$.

The critical point here: partial continuity in each variable does not guarantee full continuity. This is one of the most important distinctions in multivariable analysis.

Example: Define $f(x,y) = \frac{xy}{x^2+y^2}$ for $(x,y)\neq(0,0)$ and $f(0,0)=0$.

• Fixing $y=0$: $f(x,0) = 0$ for all $x$, so $\lim_{x\to 0}f(x,0)=0=f(0,0)$. Partially continuous in $x$.
• Fixing $x=0$: $f(0,y) = 0$ for all $y$, so $\lim_{y\to 0}f(0,y)=0=f(0,0)$. Partially continuous in $y$.
• But the multivariable limit at $(0,0)$ does not exist, so $f$ is not continuous there.

Discontinuity and Its Types

A function is discontinuous at a point when any of the three continuity conditions fails. The classification parallels single-variable types, though the geometry is richer:

• Removable discontinuity: The multivariable limit exists, but the function is either undefined at the point or defined to a value different from the limit. You can "fix" it by redefining $f(a,b)$ to equal the limit.
• Jump discontinuity: The function approaches different values along different paths. In the multivariable setting, this is often called path-dependent behavior. The $\frac{xy}{x^2+y^2}$ example at the origin is this type.
• Infinite discontinuity: $f(x,y)\to\pm\infty$ as $(x,y)\to(a,b)$. For example, $f(x,y)=\frac{1}{x^2+y^2}$ near the origin.
• Oscillating discontinuity: The function oscillates without settling on any value (finite or infinite). For example, $\sin\!\left(\frac{1}{x^2+y^2}\right)$ near the origin.