Continuity of Functions - In the Real World

In the Real World

Continuous functions, believe it or not, are all sorts of useful. For one thing, they're the secret behind digital recording, including CDs and DVDs.

Here's a brief explanation of how continuous functions are used for recording. Suppose you want to use a digital recording device to record yourself singing in the shower. The song comes out as a continuous function. The digital recording device can't record what you sound like at every moment in time (there are infinitely many moments!), but it can record little bits of what you sound like several times a second (actually, way more often than that).

Since the song is a continuous function and continuous functions are nice (in all the ways we talked about earlier and in many other ways), the several-times-a-second recording contains enough information for a computer to reproduce more-or-less what you sounded like the whole time you were singing. If the little bits were recorded frequently enough and carefully enough, the reproduction will sound just like you.

Check out how Digital Recording works.

To see how mathematical this can be, check this one out.

• Page: I Like Abstract Stuff; Why Should I Care?

  Continuous functions as we've introduced them here are just the tip of the iceberg. The field of point-set topology defines more general notions of open and closed sets and then defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function. We can have much weirder functions that don't have numbers as inputs or outputs, and yet are still continuous.

• Page: How to Solve a Math Problem

  There are three steps to solving a math problem.

  • Figure out what the problem is asking.
      
  • Solve the problem.
      
  • Check the answer.

  Example. Determine all values of x at which the function 

  

  is discontinuous.

  Answer:

  • Figure out what the problem is asking.

  We want to make sure we understand the problem. What does discontinuous mean? It means "not continuous", but what does that mean?

  A function is continuous at a value x = c if three things happen:

  • f(c) exists,
      
  •  exists, and
      
  • 

  For the function to be discontinuous at x = c, one of the three things above need to go wrong. Either

  • f(c) is undefined,
      
  • doesn't exist, or
      
  • f(c) and  both exist, but they disagree.

  This problem is asking us to examine the function f and find any places where one (or more) of the things we need for continuity go wrong.

  • Solve the problem.

  • Where is f(x) undefined?

  Since we're looking at a rational function, f is undefined wherever its denominator is 0. To find where that is, we need to factor the numerator and denominator.

  

  When the denominator is 0, either x = -3 or x = 4.

  • Where does  not exist?

  This rational function has a hole at x = -3 and a vertical asymptote at x = 4, therefore doesn't exist. This gives us another reason that f(x) is discontinuous at x = 4.

  • Where do f(c) and  both exist, but disagree?

  This function doesn't have any places like that! Since a rational function is continuous everywhere it's defined, we've found all the discontinuous places we need to worry about.

  To summarize, this function is only discontinuous at x = -3 and x = 4.

  • Check the answer.

  Besides doing the arithmetic again, probably the best thing to do is graph it with a calculator. Make sure it looks continuous except at x = 4, where there should be an asymptote. If we ask the calculator what the function is for x = -3, it should say "ERROR," because f(-3) is undefined.

• Appendix: Intervals and Interval Notation

  An interval on the real line is the set of all numbers that fall between two specified endpoints.

  Let a and b be real numbers with a < b. We can have the following types of finite intervals:

  • The open interval (a, b) is the set of all real numbers that fall strictly in between a and b. That is, all real numbers x with a < x < b. The values a and b are not included in this interval.

  • The closed interval [a, b] is the set of all real numbers x that fall (non-strictly) in between a and b. That is, all real numbers x with a ≤ x ≤ b. The values a and b are included in this interval.

  • The interval [a, b) is the set of all real numbers x with a ≤ x < b. a is included in this interval, while b is not.

  

  • The interval (a, b] is the set of all real numbers x with a < x ≤ b. We include b in this interval, but not a.

  The values a and b are called endpoints because they're the points at either end of the interval.

  To remember which of the two intervals—(a, b) or [a, b]—includes the endpoints a and b, try thinking of the interval notation like arms. If the arms are like the brackets [a,b] then they are holding a and b firmly in there. If arms are like the parentheses (a,b), then the endpoints a and b slip out.

  The interval (a, b) is called open, while the interval [a, b] is called closed. The intervals [a, b) and (a, b] are neither open nor closed. We might hear these intervals called "half-closed," "semi-closed," "half-open", or "semi-open".

  We can also have infinite intervals. Since ∞ isn't a number, we'll always have parentheses around ∞ or -∞, not closed brackets. Here are the types of infinite intervals we can have,  assuming that a is some finite number:

  • (-∞, ∞), which is the whole real line. In other words, this is the set of all real numbers.
      
  • (-∞, a) is the set of real numbers x with x < a. This interval does not include a.
      
  • (-∞, a] is the set of real numbers x with x ≤ a. This interval does include a.
      
  • (a, ∞) is the set of real numbers x with a < x. This interval does not include a.
      
  • [a, ∞) is the set of real numbers x with a ≤ x. This interval does include a.

  The infinite intervals (-∞, ∞), (-∞, a], and [a, ∞) are closed intervals. The infinite intervals (-∞, a) and (a, ∞) are open.

  The infinite intervals (-∞, ∞), (-∞, a], and [a, ∞) are closed intervals. The infinite intervals (-∞, a) and (a, ∞) are open.