The Art and Craft of Problem Solving

9.2 CONVERGENCE AND CONTINUITY 323

1. First, show that V'n + 1 - Tn = O(n-^2 /^3 ).

2. All that matters about O(n-^2 /^3 ) is that it can be made arbitrarily small for large
   enough n. So now let's try to get differences of cube roots close to a particular
   number, say, n. If we wanted to get just moderately close, we could look at the
   sequence

?fil, ?/28, m, V'3Q, ...

that begins at 3 < n and partitions the rest of the number line into intervals that

are no wider than V'28 -3. If we wanted to get even closer, we could start at

3, as before, but represented by the difference V' 103 - .j73. So now we have

the more finely spaced sequence

3, V'lOOI -7, V'1002 -7, ....

3. Make this rigorous, and general (not just for n) and you're done.

Continuity

Informally, a function is continuous if it is possible to draw its graph without lifting the
pencil. Of the many equivalent formal definitions, the following is one of the easiest
to use.

Let f: D � IR and let a E D. We say that f is continuous at a if

lim f(xn) = f(a)

n-+oo

for all sequences (xn) in D with limit a.

We call f continuous on the set D if f is continuous at all points in D.
Continuity is a condition that you probably take for granted. This is because virtu­
ally every function that you have encountered (certainly most that can be written with a
simple formula) are continuous.^5 For example, all elementary functions (finite combi­
nations of polynomials, rational functions, trig and inverse trig functions, exponential
and logarithmic functions, and radicals) are continuous at all points in their domains.
Consequently, we will concentrate on the many good properties that continuous
functions possess. Here are two extremely useful ones.

Intermediate-Value Theorem (IVT) If f is continuous on the closed inter­
val [a,b], then f assumes all values between f(a) and f(b). In other words, if y
lies between f(a) and f(b), then there exists x E [a, b] such that f(x) = y.
Extreme-Value Theorem Iff is continuous on the closed interval [a,b], then
f attains minimum and maximum values on this interval. In other words, there
exists u, v E [a,b] such that f(u) � f(x) and f(v)? f(x) for all x E [a, b].
The extreme-value theorem seems almost without content, but examine the hy­
pothesis carefully. If the domain is not a closed interval, then the conclusion can fail.

SNotable exceptions are the floor and ceiling functions LxJ and rxl -