Chapter 1: Limits.

Section 1.3 and 1.4 (More Complicated Limits and Continuity):

Continuity:

The notion of limit intentionally removes consideration of the value of the function AT a point, so that we can deal with "holes" in functions as mathematical objects in and of themselves.

At other times, however, it is important to limit our discussions to functions that have no holes, breaks, "smears," or asymptotes.  To do this, we need to "plug the holes" in our limit definition.  This brings up the notion of continuity.

A function, f(x) is called continuous at a point x = C if the following conditions hold:

1.  "C" is in the domain of f(x).  That is, f(C) exists.
2.  f(x) is defined in an interval containing C (C can't be on the "edge" of a domain.)
3.  The lim_x->Cf(x) exists, which means that the limits from the left and right both exist and "match up."
4.  The limit, lim _x->C f(x) = f(C).  (The point (C, f(C) ) is where it "belongs," so that there's no "hole.")

(This definition is redundant in several respects, but it emphasizes what you need to look for to see if a function is continuous at a point.)

Here are the basic ways that a function can fail to be continuous:

1.  "Holes," or "removable discontinuities":

ex. f(x) = (x²- 4)/(x + 2) , f(-2) = -2:

2.  "Jump" discontinuities:

ex:  f(x) = abs(x² - 4)/(x+2)

3.  Asymptote Discontinuities:

ex) f(x) = (x² +  4)/(x + 2)

4.  "Strange" Discontinuities:

ex) f(x) = sin(1/(x+2))

5.  "Edge" discontinuities:

ex) f(x) = sqrt(x + 2)

This last example points out a new definition that is often useful:

If a point is on the "edge" of a domain, we can talk about "one-sided continuity."

f(x) is "continuous from the right" at x = A if

1.  "A" is in the domain of f(x).
2.  f(x) is defined in an interval to the right of "A." (Think of "A" as being on the left edge of your interval.)
3.  lim_x->A+ f(x) = f(A)  (The limit from the right exists and equals f(A).)

A similar definition exists for "continuous from the left at x = B.

Finally, a function f(x) is called "continuous on the closed interval, [A, B]" if
1. f(x) is continuous for every point "C" in the interior of the interval, (A, B)
2. f(x) is continuous from the right at x = A.
3. f(x) is continuous from the left at x = B.

Functions that are continuous on closed intervals play a very important role in mathematics.  Physically, they are something like taking a piece of string, and tacking it down on both its left and right ends.  Because they are "tied down" and have no "breaks" or "holes," they have many very important properties, most of which follow from the fact that the real numbers themselves form a "topologically complete" system; that is, they contain no "holes."

The most important result for our (calculus) purposes from real number topology is the following:

The continuous image of a closed interval is either a closed interval or a single point.

This fact is the basis for two big theorems, called the "Intermediate Value Theorem," and the "Existence of Extrema" theorem (also known as the non-calculus Max-Min Theorem.)

The first of these says that a continuous function cannot get from (A, f(A)) to (B, f(B)) without passing through all of the heights between f(A) and f(B).  We'll look at that one in the applet below.
 

The second says that continuous functions on closed intervals "achieve optimality;" that is, there is some point, c_M on [A, B] such that f(c_M) = M >= f(x) for all other x in [A, B], and similarly for the minimum value.  We'll look at that one in Chapter 3.

Here's an applet demonstrating both the "continuous image" idea, as well as the IVT and the Extreme Value Theorem.

Applet:

Intermediate Value Theorem:  1) Let f(x) be continuous on [A, B] with f(A) < f(B).  Then, for any C, with f(A) < C < f(B), there exists at least one "E" in [A, B], with f(E) = C.

2) Let f(x) be continuous on [A, B] with f(B) < f(A).  Then, for any C, with f(B) < C < f(A), there exists at least one "E" in [A, B], with f(E) = C.

Notice that this theorem can fail if f(x) is NOT continuous on the closed interval, and it can also fail if the domain of f(x) is not "topologically complete;" for example, if the domain is the rational numbers:

Example 1: f(x) = abs(x² - 4) / (x + 2) on the closed interval [-3, 3] (see the picture above.)  Even though f(-3) = -5 and f(3) = 1, there's no point, E, in [-3, 3] with f(E) = -2.

Example 2: f(x) = x² - 2, on the interval [1, 2], if our domain is only rational numbers on [1, 2].  Even though f(1) = -1, and f(2) = 2, there is no rational number, E, in [1, 2] with f(E) = 0.  (The value we'd need is sqrt(2), which is not rational.  f(x) sort of slips through a "hole" in the rational x axis at sqrt(2).  The real x axis has no such holes.

Fact:  The most common application of the IVT is when looking for roots (zeros) of a continuous function.  Then it says that if a continuous function f(x) changes sign between x = A and x = B, it must have a root in the interval (A, B).  This is the starting point for many methods of numerically approximating a solution to an equation.

Example 3:  Show that f(x) = 5x⁷ + 3x⁵ + 2x² - 1 has a root between x = 0 and x = 1.

Solution, since f(x) is continuous (it's a polynomial!) and f(0) = -1, while f(1) = 9, f(x) must have a root in (0, 1).

You can then "zoom in" on the root by using the method of bisection (check out f(1/2) to see if the root is in the left or right half of the interval, then continue to bisect and check.)

Next, some worked examples from section 1.4.