Understanding Continuity (Explaining Calculus #4)

In both life and academic science, we often come across things that change incrementally over time. If we are buying a fast car, we want to know how quickly that car accelerates from 0 to 60 miles per hour – and the answer is most certainly not zero seconds – we first have to hit every number between 0 and 60. Just like many other situations we encounter in day to day life and in more academic areas, we don’t “leap” from one point to another suddenly without first crossing over everything in between the two points.

Now, what do we make of this mathematically? How might we express these ideas of continual change over time in mathematical terms? The answers, as it turns out, lies in the ideas of limits we have developed so far.

Understanding “Continuity”

As in much of mathematics, we will phrase the idea of input-output type relationships as functions. In the car example from earlier, we might call out input the amount of time we’ve been slamming the gas pedal and the output as our current speed. In mathematics, we have a name for the idea of continual development of a function over time – we call these continuous.

How then do we define continuity? I think the most helpful way to view this is by thinking about what it would mean to be not continuous – usually called discontinuous. Imagine that the function f(x) is not continuous at 7. This would mean that, at 7, things jump in some way. Another way we could think of the same thing is that one part of the graph is ripped apart from another. Before we move on, let’s use two numerically helpful examples.

Example 1: Define the function f(x) to be x rounded down to the nearest whole number. Let’s think about what happens to x near 2. If you pick any number a that satisfies 1 < a < 2, then f(a) = 1. If you instead choose a number b that satisfies 2 < b < 3, then f(b) = 2. Now, suppose I tell you that c is very close to 2 and ask you what value f(c) has. You can’t really tell me anything, because there is a huge difference between cases when c > 2 and when c < 2.

In terms of limits, we can say here that \lim\limits_{x \to 2} f(x) does not exist, since the “left-hand side” and the “right-hand side” of f(x) look extremely different.

Example 2: This example is different than the first one in a key way. We now define f(x) differently. Define f(x) = 0 whenever x \not = 0 and f(0) = 1. Unlike in Example 1, the left and right sides of 0 are actually similar – in the language of limits, we have \lim\limits_{x \to 0} f(x) = 0. But, the value of the limit is not the same as the actual value of f at 0. It is as if we poked a hole at 0 and shifted that point up without changing anything else.

In this case, we still have discontinuity. There is a “microscopic jump” that happens at 0.

A Precise Definition of Continuity

How do we use these examples of jumps to make a more careful, mathematical definition of continuity? In other words, how do we define what it means to not have a discontinuous jump? Here is the standard mathematical definition, which we have now motivated by discussing examples where the idea breaks down.

Definition: A function f(x) is continuous at the point a whenever we have

\lim\limits_{x \to a} f(x) = f(a).

Example 1 was an example of a situation where the expression \lim\limits_{x \to a} f(x) does not make sense. Example 2 was a situation where both f(a) and \lim\limits_{x \to a} f(x) make sense. You could, if you’d like, use an example like f(x) = \dfrac{1}{x} as an example where f(a) doesn’t make sense (in this case, f(0) doesn’t make sense).

Most functions that we deal with are continuous, or at least continuous everywhere that we care about. So remembering this definition verbatim is not necessarily important unless you have to deal with an especially difficult function. In conceptual terms, the idea of continuity should be thought of as function whose graph has no rips, jumps, or holes.

Real-World Meaningfulness of Continuity

The idea of “no rips, jumps, or holes” can be reframed in a way that has obvious real-world implications. If a function f(x) is continuous, it means that if we change x by a small amount, then the output value f(x) also changes by only a small amount. What this ends up implying is that continuous functions are especially easy to approximate, which is useful in computer science and engineering – and is one of the facts that allows us to use calculators to find decimal expansions for complicated numbers and functions.

Here is an example of how that might work. Let’s say you want to approximate the value of \sqrt{3}. We can do this by using the continuous function f(x) = x^2. Since this function is continuous, then “y is close to \sqrt{3}” and “y^2 is close to 3″ are basically the same statement. You could then find a fraction whose square root you know how to compute, and if that fraction is close to 3, then its square root is very close to \sqrt{3}. Here is an example. The number 2.89 = \dfrac{289}{100} is pretty close to 3, and its square root is exactly 1.7 = \dfrac{17}{10}. This means, since f(x) = x^2 is continuous, that 1.7 \approx \sqrt{3}. This is just one example of a way that knowing about continuity can be helpful.

An Important Result of Continuity

Continuous functions are very helpful in the real world, but they are also massively important conceptually. There are so many reasons for why, most of which I won’t get into. But I will lay out two of them. The first, called the Intermediate Value Theorem, can be used to guarantee that lots of equations definitely have solutions, even if you don’t know how to write down what that solution is. The second, called the Extreme Value Theorem, guarantees that under a very general situations, questions about maximum and minimum values always make sense.

The Intermediate Value Theorem (IVT) basically says “If f(x) is a continuous function and hits the numbers a and b on its graph, then it hits every number between a and b too. The Extreme Value Theorem (EVT) basically says “If f(x) is a continuous function, then it always has a maximum and minimum value in any ‘finite window’.”

Here are more formal ways of expressing these two ideas.

Intermediate Value Theorem: Suppose that f(x) is a continuous function and that f(a), f(b) are two unequal numbers. Then if M is a number between f(a) and f(b), then f(x) = M has a solution with x between a and b.

Extreme Value Theorem: Suppose that f(x) is a continuous function. Then if f(x) makes sense for all x satisfying a \leq x \leq b, then there are both a largest value and smallest value of f(x) for x satisfying a \leq x \leq b.

It isn’t important that we go too deep into exactly why these work (but perhaps the visual ways I’ve explained continuity might suggest to you why both of these must be true). But because of the IVT and EVT, continuity lets us know that many real-world functions that matter either definitely have maximum or minimum values (which matters a great deal if that function counts how much money your company makes) or that certain equations definitely have solutions (which matters a great deal if that equation tells you something about the structural integrity of a building).

Continuity is an extremely important concept in calculus, and is one that is worth reviewing over and over again until you really deeply understand what it means. Going forward, pretty much every situation we find ourselves in will make use of continuous functions, so this is good to keep in mind.

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