Continuity is one of the most important concepts in all of mathematics. Without continuity, we wouldn't have calculus.
Without calculus, we wouldn't have the computer nor the internet.
And of course, no internet, no Elevri. How sad right?
So we should really be thankful for continuous functions, as they are crucial to the modern world.
Back in single variable calculus, continuity was at a point was easy to check. All we had to do was to check if the function value at that point coincided with the limit as we approached the point from the left and the right.
But now in multivariable calculus, there are more ways to approach a point, since we are working in higher dimensions.
Therefore, we have to check that the function value at a point coincide with the limit as we apporach from all directions.
For a function to be continuous at a point , need to satisfy
It is easier to prove that a function is not continuous at a point than the other way around. Thankfully, there are a few techniques we can use, such as
Comparing with known limits using squeeze theorem
Building up towards limits
The concept of limits in multivariable calculus is just the multidimensional friend of limits in one variable. However, this time we'll be more rigorous, so brace yourself and buckle up for the ride. We'll stay with functions of two variables, but the definition generalizes to higher dimensions.
First, we introduce what we call a neighborhood.
A neighborhood of a point in is a set:
for some .
In the -plane, the neighborhood of a point is a small disc around the point.
Definition of a limit
The definition of a limit in multivariable calculus has two parts.
We say that
if these two conditions hold:
every neighborhood of contains points in the domain of that are not itself
for every number , there is a number such that holds for each in the domain of , and satisfies .
We've here introduced the two very small beasts called and . They appear in many definitions with the same meaning as here, so read this carefully. We'll go through part of the definition step by step.
We start off by creating the two variables and .
Next, we choose a region around the limit on the -axis, . This region forms a flat sandwich with bread and as spread.
Then, using the region, we can define a region with as its radius in the -plane, around the point . This region needs to be such that all the values including inside the region are mapped to -values in the region.
The final step is saying that we can find -values arbitrarily close to the value , by using only points lying inside a sufficiently small region centered around .
Now, take a breath and re-read the definition.
Parallel to limits in one variable
Now, some practical aspects. Recall how we in single variable calculus said that the limit existed only if the right and left limit were the same. The second part of the definition implies that now that we are in three dimensions, in order for the limit to exist it must be the same no matter how we choose to approach the point.
The limit exists only if walking along any curve in the domain of the function towards the point, we always end up at .
The graph below shows a plane with an infinitely thin (one-dimensional) line cut out. Approaching the point along that line, we have no problem.
However, trying to approach along a line parallel to the -axis, we will not find as goes to , as the line is infinitely thin: we had to start the journey in the lower plane, and we can't just jump up to the line from there. Thus, the limit at does not exist.
Finally, these are some laws for combining limit.
Say we have
and every neighborhood of the union of the domains of and contain points other than . Then,
and if is continuous at t = L
Calculating multivariable limits
As discussed when we were introduced to multivariable limits,
only if approaches as approaches along any curve in the domain of the function.
Now the problem with this is that there may be infinitely many such curves.
As opposed to limits in one variable, where we only had to check the two cases where we approach a point from above and from below, there is no guarantee that we can split the process into a finite number of cases.
In light of the possibly infinite number of curves, if we have reasons to believe that the limit does not exist, we usually start there.
All it takes is two curves for which approach different values as we walk along them toward a point, and no limit will exist in the point.
Alternatively, one curve along which approaching the point does not result in a finite value for
would also suffice.
We want to evaluate the following limit:
Checking the limit along all the lines :
Checking the limit along the curve :
Hence, the limit does not exist.
If we fail to prove that the limit does not exist, we unfortunately don't have a universal method that will always work to find the limit. That being said, there are some standard procedures:
In certain cases, plugging in for and for can right away yield the limit we are after.
As an example:
Sometimes, we can go from a multivariable limit to a single variable one by making a variable substitution.
Consider this limit:
where we can swap a for :
This is now a limit in only one variable, with the quotient tending to as , and we can solve it using L'Hopital's rule:
Some limits are more readily evaluated if we first convert the representation of the point from Cartesian coordinates to polar form .
Now since has disappeared from the expression, it can take on any value for any , and so the limit does not exist!
The squeeze theorem is not restricted to functions of one variable, and so another method that might help us is to close in the expression between two limits that we know to be equal.
Have a look at this example:
where we can use the squeeze theorem to calculate the limit as:
Thus, the limit is .
Using the definition
One last technique we can employ is to ride on the definition of a multivariable limit. With the definition being rather technical, this method is a bit involved. Do not worry about not seeing it clearly when reading the example for the first time.
This will be used to prove that
Since we have:
Similarly, by :
Combining them, we get:
Now let be any positive number, and set . Then by the inequality above
which by the definition proves that the multivariable limit is .
Mathematicians are obsessed with the idea of continuity. It's almost as if there was a cult of continuity. See, continuous functions have many desirable properties.
Roughly speaking, a multivariate function is continuous if the graph looks like an elastic piece of cloth. No holes, no surprises.
A function is continuous at a point if the limit coincides with the function value.
This means that
And we say a function is continuous as a whole if it's continuous in its entire domain.
Here's a fun fact: if you toy around with two continuous functions, adding, subtracting, multiplying, dividing, the result will be a continuous function.