Real Analysis 177
The process of estimating errors is fundamental in real analysis, and the analyst’s attitude
can be expressed with the slogan:
In algebra, it is equal, equal, equal.
But in analysis, it is less-or-equal, less-or-equal, less-or-equal.
The point is that, when proving, say, that a function is continuous, you have to bound the
size of |f (x) f (c)|, to keep it less than , and the way that you do this is by showing that
it is less than or equal to one thing, which is less than or equal to another, and so on, until
you can get as the final bound. This process is called estimation.
The reader will show in exercise 15.1 how theorems 132 and 133 can be used to show
that every polynomial function on the real numbers is continuous.
15.3 Continuous at exactly one point
We all know how a collection of good examples can illuminate a mathematical idea, but
often having some good nonexamples can also be enlightening, by helping one to see a
fuller range of possibilities. So let us consider the following crazy function, which I claim
is continuous at exactly one point only and discontinuous at all other points.
0
f (x) =
x
2
x Q
0 x Q
The function is defined essentially as the union of two pieces, by splitting the domain
into the rational numbers and the irrational numbers, and doing one thing on rational values
and another on irrational values. Specifically, the function f (x) agrees with x
2
when x is
rational (shown in blue), but otherwise, it is 0 (shown in red). Despite this bifurcation
of cases, the function is actually continuous at 0, since the two separate functions come
together at that point. The function is continuous at 0 because for any >0, we can ensure
that f (x) is within of f (0) = 0 by taking x within δ =
of 0. Meanwhile, the function
is discontinuous at all other points c 0, since some values of x near c will be very near
c
2
, but other values of x near c will be 0, and so they cannot all be within of f (c)if is
smaller than c
2
.