What is the Definition of Continuity Calculus

As stated in the previous lesson, the continuity chapter has opened up. Even though the last chapter was when the Calculus lessons began, that was like midnight (when the date changes), as all of the other early lessons are in the depths of the night (10:00 PM to 3:59 AM) on the AM side. Now it's dawn, the last phase of night before the sun starts rising. The last chapter was all about how to evaluate limits. This chapter is about what you can do with limits, which also includes the foundations of one of the core concepts of Calculus – derivatives. The sun starts rising when the differentiation lessons begin.

So what is continuity? Going over the definition of a function, is the set of all points that follow the expression. And when I mean all, what does that mean?

• If I'm referring to just natural numbers, only positive integers are considered. For instance, if , what value of will yield . The answer is none since is indivisible by . Even and negative numbers won't count since it's considering natural numbers only. So this function is a set of countable points, not a line.
• If I'm referring to integers only, the function is still discrete, and thus cannot be represented by a line.
• If I'm referring to rational numbers only, the function may consider all rational numbers, no matter how precise they are, but not irrational numbers. So you can find a value for at when , but not when . So the function cannot be represented by a line, but rather a series of points.

This is what I mean by all points. It means all real numbers, including the irrational numbers. If irrational numbers aren't to be considered, how many numbers are between ? Just 11 points (if you count 0), or even more, if you consider rational numbers with a single-digit denominator. How many numbers are between if irrational numbers are considered? The answer is infinite. If all points can be considered, then you can have a line. All points that lie on the line are part of the function, and if a point on the line is part of the function, you can travel along. You will have to stop if a point on the line is not part of a function. Not stopping means you're continuing.

Think of it as a year. At what point of the year is time continuous, using seconds as the smallest unit and months as the largest unit?

• If you consider months only, then time is not continuous in the year, not even by day. If it is by day, you would only consider the first day of the month.
• If you consider weeks and months only, then time is not continuous in the year. If it is by day, you would only count one day of the week only.
• If you consider days, weeks, and months only, then time is not continuous in the year. The only time of the day that would matter is midnight.
• If you consider hours and greater only, then time is not continuous in the year. You would count every hour, but only the minute the hour begins.
• If you consider all, except for seconds, then time is not continuous in the year. You would consider every minute, but only the second the minute begins.
• If you consider all, including seconds, then time is continuous in the year.

When you first learn how to count, you would learn natural numbers only. Like let's say you have 10 balls on the floor. Natural numbers would help indicate how many balls you have. But integers would indicate the change in number of balls, rational numbers indicate parts of the ball's surface area, and irrational numbers fill up the entire surface area of the ball. That's how continuity works.

Definition of Continuity

The definition of continuity states that a function is continuous at a point if the limit as x approaches the point equals the function evaluated at the point. In other words:

In order for a function to be continuous at a point, it must have three things in common:

If even one of these conditions fail, then the function is not continuous at the point.

Let's take . Is the function continuous at any of these points?

Starting with the first one, we can evaluate the limit of the function algebraically. factors into , whereas factors into . Reducing the fraction, you're left with . At , the limit equals . Even if we evaluated the function numerically, is equal to . Unfortunately, if we evaluate the original function at , it will result in zero-division, invalidating the function. So is not continuous on .

Going to the next one, given that the function isn't a piecewise function (proving that equals ) and that limits exist at a point no domain restriction exists (proving that exists), all we need to prove is that if is defined on . , making the denominator , therefore it's defined on . Since all three properties hold, is continuous on .

Now let's look at the last one. Even if we reduce the function to , you still get an undefined result at , meaning that neither is the function defined, nor does the limit exist. It's also worth mentioning that and , failing the last test. So this function isn't continuous on .

There's also the definition of left-continuous and right-continuous. A function is left-continuous if these properties hold:

And it's right-continuous if these properties hold:

If it's both left-continuous and right-continuous, then the function is continuous at that point. On many functions, it's easy to pass the continuity test on many points, so most of the failures on the left-continuous and right-continuous test come from piecewise functions.

Anyway, that's it for the lesson. The rest of the chapter is going to continue the properties of continuity, which also includes foundations for differentiation.

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Source: https://snailpacedcalculus.com/2022/07/25/math-lesson-21-definition-of-continuity/

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