Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 715
domain, we equivalently define continuity in terms of the Euclidean distance
between two points in the domain and the Euclidean distance between the
corresponding function values.
There are various types of continuity, and two functions help to illustrate
the differences.
Definition 0.1.30 (continuous function)
Let f be a real-valued function whose domain is a set D ⊆ IR d . We say that
f is continuous at the point x ∈ D if, given ɛ > 0, ∃δ ∋ ∀y ∈ D ∋ x − y <
δ, f(x) − f(y) < ɛ.
Here, the norms are the Euclidean norms. Notice that the order of f(x)
may be different from the order of x.
The δ in the definition may depend on x as well as on ɛ.
If f is continuous at each point in a subset of its domain, we say it is
continuous on that subset. If f is continuous at each point in its domain, we
say that f is continuous.
We have an immediate useful fact about continuous functions:
Theorem 0.1.14
If f is a continuous function, the inverse image f −1 of an open set is open.
Proof. Follows immediately from the definition.
Example 0.1.6 (the Dirichlet function;) nowhere continuous function
The indicator function of the rational numbers, called the Dirichlet function,
is everywhere discontinuous.
Example 0.1.7 (the Thomae function;) continuous on irrationals, discontinuous
on rationals
Let f(x) be defined as
⎧
1 if x = 0
⎪⎨ 1 p
if x = is rational,
f(x) = q q
⎪⎩
where q is a positive integer and p is relatively prime to q
0 if x is irrational
Then f(x), called the Thomae function, is continuous at x if x is irrational
and discontinuous at x if x is rational.
We now consider three successively stronger types of continuity.
Definition 0.1.31 (uniformly continuous function)
Let f be a real-valued function whose domain includes a set D ⊆ IR d . We say
that f is uniformly continuous over D if, given ɛ > 0, ∃δ ∋ ∀x, y ∈ D with
x − y < δ,
f(x) − f(y) < ɛ. (0.1.28)
Theory of Statistics c○2000–2013 James E. Gentle