2.3 Limits and continuity in multiple variables

Functions of several variables expand our understanding of limits and continuity. We now explore how these concepts apply to functions with multiple inputs, introducing new challenges and considerations in evaluating limits and determining continuity.

Directional limits and path dependence become crucial in multivariable functions. We'll learn how to assess continuity by examining limits from different approaches and understand why some functions may be continuous in individual variables but not overall.

Limits of Multivariable Functions

Evaluating Limits and Directional Limits

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• Limit of a multivariable function $[f(x,y)](https://www.fiveableKeyTerm:f(x,y))$ as $(x,y)$ approaches a point $(a,b)$ is denoted as $\lim_{(x,y)\to(a,b)}f(x,y)=L$
  • If the limit exists, the function approaches the value $L$ as the point $(x,y)$ gets closer to $(a,b)$ from any direction
• Directional limit is the limit of a function as $(x,y)$ approaches $(a,b)$ along a specific path or direction
  • Different paths to the same point can yield different limit values (path dependence)
  • Example: $\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+y^2}$ has different limits along the paths $y=mx$ and $y=x^2$
• Path limit is the limit of a function along a specific path or curve as the parameter approaches a certain value
  • Parametric equations define the path, and the limit is evaluated as the parameter tends to a specific value
  • Example: $\lim_{t\to0}f(t\cos t,t\sin t)$ is a path limit along the spiral $x=t\cos t, y=t\sin t$ as $t\to0$

Formal Definition and Iterated Limits

• $\varepsilon-\delta$ definition of limit for multivariable functions: $\lim_{(x,y)\to(a,b)}f(x,y)=L$ if for every $\varepsilon>0$, there exists a $\delta>0$ such that $|f(x,y)-L|<\varepsilon$ whenever $0<\sqrt{(x-a)^2+(y-b)^2}<\delta$
  • This definition formalizes the idea that the function values get arbitrarily close to $L$ as $(x,y)$ approaches $(a,b)$
• Iterated limit is the process of evaluating limits one variable at a time, fixing the other variable(s)
  • $\lim_{x\to a}\left(\lim_{y\to b}f(x,y)\right)$ and $\lim_{y\to b}\left(\lim_{x\to a}f(x,y)\right)$ are iterated limits
  • If both iterated limits exist and are equal, the limit of the function exists; otherwise, the limit may not exist or be path-dependent
  • Example: $\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+y^2}$ has different iterated limits, so the limit does not exist

Continuity in Multiple Variables

Continuity and Partial Continuity

• Continuity in multiple variables: A function $f(x,y)$ is continuous at a point $(a,b)$ if $\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)$
  • The function value at $(a,b)$ must equal the limit of the function as $(x,y)$ approaches $(a,b)$
  • Example: $f(x,y)=x^2+y^2$ is continuous at every point in its domain
• Partial continuity: A function is partially continuous with respect to a variable if it is continuous when treating the other variable(s) as constant
  • $f(x,y)$ is partially continuous in $x$ at $(a,b)$ if $\lim_{x\to a}f(x,b)=f(a,b)$
  • $f(x,y)$ is partially continuous in $y$ at $(a,b)$ if $\lim_{y\to b}f(a,y)=f(a,b)$
  • Example: $f(x,y)=\frac{xy}{x^2+y^2}$ is partially continuous in $x$ and $y$ at $(0,0)$, but not continuous at $(0,0)$

Discontinuity and Its Types

• Discontinuity in multiple variables occurs when a function is not continuous at a point
  • The limit of the function may not exist, or the limit may not equal the function value at that point
  • Example: $f(x,y)=\frac{xy}{x^2+y^2}$ is discontinuous at $(0,0)$ because the limit does not exist
• Types of discontinuities in multiple variables are similar to those in single-variable functions
  • Removable discontinuity: The limit exists, but the function is not defined or has a different value at the point
  • Jump discontinuity: The function has different limits as $(x,y)$ approaches the point from different directions
  • Infinite discontinuity: The limit of the function is infinite as $(x,y)$ approaches the point
  • Oscillating discontinuity: The function oscillates without approaching a specific value as $(x,y)$ approaches the point