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05 September 2003

Continuity

by George E. Hrabovsky, President, MAST

Where We Have Been

Last week we completed our discussion of limits.  We have not exhausted the subject of limits by any stretch of the imagination.  Limits are at the heart of calculus, but they are also the basis for differential geometry and topology.

In this column we will discuss a class of function that allows us to use all of the tools we have been working on up to now.  We are nearly to the point where we can begin going back to connecting the mathematics to real life.

The Definition of Continuity

If I tell you that the property of continuity is important, you will have to take my word for it.  Its importance stems from subjects that we have not yet discussed, so its importance is not yet apparent.  When calculus was first developed, the notion of continuity was developed only after calculus was established.  For now, you will just have to take my word for it.

If the following is true,

Underscript[lim, xx_0] f(x) = f(x_0),

then f(x) is said to be continuous at x_0.

Three Theorems

This first theorem establishes the basic arithmetic of continuous functions.

Theorem 1

If the functions f, g, h, ... are continuous at x_0, then so are f ± g ± h ± ... and so are f g h ....

The second theorem establishes the division properties of continuous functions.

Theorem 2

If the functions f and g are continuous at x_0 and g≠0, then so is f/g .

The final theorem discusses composite functions.

Theorem 3

If the functions g is continuous at x_0 and f is continuous at g (x_0), then so is f∘g .

The Math Challenge from Last Time

The task was to prove the two theorems about limits at infinity.  The first was that the function f(x) L as x +∞ if and only if there exists another function f^*(ξ) = f(1/ξ) L as ξ0 from the right.  We also have f(x) L as x -∞ if and only if there exists another function f^*(ξ) = f(1/ξ) L as ξ0 from the left.  We shall proceed to prove this by cases.  There are five cases; x +∞, x -∞, L = +∞, L = ∞, and L = -∞ .

We begin with the case x +∞.  If L≠ ∞, ± ∞ then for any ϵ>0 there exists an arbitrarily large number M>0 such that | f(x) - L | <ϵ if x > M.  However, in terms of the new variable, ξ = 1/x, we have,

| f(x) - L | = | f(1/1/x) - L | = | f * (1/x) - L | = | f^*(ξ) - L | .

If we choose δ = 1/M we see that 0<ξ<δ implies that x>M, and thus | f^* (ξ) - L | < ϵ.  Since we have decided not to specify what ϵ is, it is considered arbitrary.  It then follows that f^* (ξ) L as ξ0 from the right.

If x -∞, and if L≠ ∞, ± ∞ then for any ϵ>0 there exists an arbitrarily large number -M<0 such that | f(x) - L | >ϵ if M > x.  However, in terms of the new variable, ξ = 1/x, we have,

| f(x) - L | = | f(1/1/x) - L | = | f * (1/x) - L | = | f^*(ξ) - L | .

If we choose δ = -M we see that 0<ξ<δ implies that M>x, and thus | f^* (ξ) - L | > ϵ.  Again we have decided not to specify what ϵ is, so it is considered arbitrary.  It then follows that f^* (ξ) L as ξ0 from the left.
    If L = +∞, then for any arbitrarily large number M>0, there exists another large number N such that when f(x) >M if x>N.  Therefore we have f^*(ξ) >M if 0<ξ<δ = 1/N.  From this, it follows that f^* (ξ)  +∞ as ξ0 from the right.

If L = -∞, then for any arbitrarily small number -M<0, there exists another small number N such that when f(x) >M if x>N.  Therefore we have f^*(ξ) >M if 0<ξ<δ = 1/N.  From this, it follows that f^* (ξ)  -∞ as ξ0 from the left.

Finally, if L = ∞,  this is just a generalization of the two previous possibilities.  Thus the theorem is proven.

The second theorem to prove was if the functions f(x) L and g(x) L ' as x +∞ then f(x) ± g(x)  L ± L '.  For any ϵ>0 we can choose some positive large numbers, M and N, such that | f(x) - L | >ϵ/2 if x>M and | g(x) - L ' | >ϵ/2 if x>N.  Then

| f(x) ± g(x) - (L ± L ') | ≤ | f(x) ± g(x) | + | L ± L ' | <ϵ/2 + ϵ/2 = ϵ

if x>max {M, N} (in other words, f(x) ± g(x) L ± L ' as x +∞.

The Math Challenge

Can you prove the theorems of this column?

Math Resources for Limits

Online:

http://www.wikipedia.org/wiki/Continuity_(mathematics)

This is a rather interesting page, Wikipedia is a free online encyclopedia.  For a more challenging page look at this one:

http://www.maths.qmw.ac.uk/~phk/AFS5.pdf

bear in mind that it is a PDF and requires Acrobat or Ghostview (or other viewers appropriate to your OS).

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a very nice chapter on limits that includes a very detailed discussion of limits at infinity.

Created by Mathematica  (September 4, 2003)