Limit and continuity of a multivariable function

The basic concept remains the same. However, with 2D things are more complicated than in 1D. In 1D you either walk forwards or backwards. In 2D we can circle around a point, meaning that, sometimes, the limit may not exist in one direction while it does exist in another. In 3D think about optical illusions. An object may appear continuous from one angle, yet it's discontinuous from another. One such example is the Penrose triangle.

For one variable we take one step to the right or one step to the left. For two variables we can take one step up or down, in addition to left and right. If we can take one small step in any direction on a plane what we are describing is a circle. Going to the 3D space and we have a sphere.

We have an equation of a circle in 2D and the equation of a sphere in 3D. The equation for the circle is: [math]\displaystyle{ (x - x_0)^2 + (y - y_0)^2 = \delta^2 \iff \delta = \sqrt{(x - x_0)^2 + (y - y_0)^2} }[/math] (we aren't interested in a negative radius, we can disregard the negative root). Suppose that [math]\displaystyle{ P = (x,y) }[/math] is located anywhere in that circle, excluding the circle's perimeter. If you know how to calculate the distance between two points from analytical geometry you are going to notice that we just wrote it. With [math]\displaystyle{ \delta }[/math] being the radius, [math]\displaystyle{ (x_0, y_0) }[/math] the circle's origin and [math]\displaystyle{ P = (x,y) }[/math] any point inside that circle.

Notice how the figure is also a graphical depiction of the property: [math]\displaystyle{ |a - b| = \sqrt{(a - b)^2} }[/math]. Distance cannot be negative. We can view the coordinates of the points as displacement vectors. Both points being displaced from the origin of the Cartesian plane to their positions shown in the graph (sum a vector with a point). The radius of that circle can also be interpreted as [math]\displaystyle{ ||\overrightarrow{P} - \overrightarrow{C}|| = \delta }[/math], where [math]\displaystyle{ \overrightarrow{C} }[/math] is displacement vector for the circle's origin.

Notation is really the same idea from limits of single variable functions:

[math]\displaystyle{ \lim_{(x,y) \to (x_0,y_0)} f(x,y) = L }[/math] (same for any number of variables)

For each [math]\displaystyle{ \epsilon \gt 0 }[/math], there is a [math]\displaystyle{ \delta \gt 0 }[/math], such that every [math]\displaystyle{ (x,y) \in D_f }[/math], [math]\displaystyle{ 0 \lt \sqrt{(x - x_0)^2 + (y - y_0)^2} \lt \delta \implies |f(x,y) - L| \lt \epsilon }[/math]. (some textbooks replace the square root with a norm and difference between the coordinates, it's really the same thing)

The concept is virtually the same used for single variable functions. We are considering the smallest distance between two points in 2D that is as close as possible to zero. While the error, the distance between the image and the limit, is the lowest possible value. Note that the definition of a limit for many variables is not considering the path to the point. The concept of one sided limits for many variables is a bit more complicated because when we expand to 2D and 3D there is up and down, front and back, there are way more sides and directions to account for.

Terminology: when a limit does exist at a point [math]\displaystyle{ (x_0, y_0) }[/math], that point is called a limit point or an accumulation point. That term comes from topology.

Continuity: the discussion is exactly the same for one or many variables.

[math]\displaystyle{ \lim_{(x,y) \to (x_0, y_0)} f(x,y) = f(x_0, y_0) }[/math]

If the function is defined at [math]\displaystyle{ (x_0, y_0) }[/math] and the limit converges to that point, the function is continuous at that point. Extend the same reasoning to any arbitrary [math]\displaystyle{ (x, y) }[/math] point chosen from the function's domain and the function is continuous everywhere in its domain.

Squeeze theorem for many variables

It's the same concept from single variable functions, there is no difference.

If [math]\displaystyle{ f(x,\ y) \leq g(x,\ y) \leq h(x,\ y) }[/math] for [math]\displaystyle{ 0 \leq \sqrt{(x - x_0)^2 + (y - y_0)^2} \lt \delta }[/math] and

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} f(x,y) = L = \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} h(x,\ y) }[/math]

Then

[math]\displaystyle{ \lim_{(x,\ y) \ \to \ (x_0,\ y_0)} g(x,\ y) = L }[/math]

The same property of limits for one variable applies to many variables. If a bounded function is multiplied by another with a limit that goes to zero, we can say that the limit of the product is zero too.

"Multi-sided" limits

When we have a limit of a single variable function there is only left and right. We have an [math]\displaystyle{ x }[/math] and we calculate the limit by considering [math]\displaystyle{ x \pm a }[/math], where [math]\displaystyle{ a }[/math] is a very small quantity. Let's try the same reasoning for two variables. Now we have two variables, [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. There is left and right, [math]\displaystyle{ x \pm a }[/math]. As well as up and down, [math]\displaystyle{ y \pm a }[/math]. If we could rely on the axis alone we'd have up to four different ways to approach a point, up x down x left x right. However, we'd be forgetting all the space in between the axis. We can circle around a point, which would mean at least 360 different paths towards a single point according to the unit circle. And even more paths for three and more variables. It's impractical to calculate a limit a hundred times just to check whether it exists or not. We need something to deal with this scenario.

It turns out that we have to resort to the knowledge of trajectories, parametric equations. When we plot level curves for functions of two variables we have a path and an equation with two variables, under the special condition that the equation always keep the level a constant. What we are looking for isn't really a vector valued function that describes a trajectory in 3D, the path over the surface of the two variable function. Our problem is that we have a starting point anywhere on the function's domain, the XY plane or part of it, and we wish to walk towards another point, the point that we wish to check whether the limit exists or not. It's hard to picture it, but every step we take in any direction on the function's domain, is reflected on the function's graph.

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Which are the easiest paths to check? The XY axis are the most obvious ones. We keep one variable equal to zero and reduce the limit of two variables to a single variable. The next one is to make one variable equal to the other [math]\displaystyle{ x = y }[/math], which translates to walking over the diagonal of the XY plane.

From a certain perspective we are using a technique that is looking at the problem of a limit of a function of two variables by imposing certain conditions that allow us to treat it as a limit of a single variable function. When we make [math]\displaystyle{ x = 0 }[/math] or [math]\displaystyle{ y = 0 }[/math] we aren't really transforming a function of two variables into a single variable one. What we are doing is "slicing" the function with a plane and looking at its "silhouette" left by intersecting it with that plane. That silhouette can be interpreted as a single variable function because it's really a line with zero thickness.

We can very well extend the same reasoning to three and more variables. The textbooks that I know don't have exercises of limits for more than two variables. I suppose it's just an exhausting process to consider even more directions and paths in 3D and beyond.