236 Chapter 5 • Continuous Functions
- Open Set Definition of Continuous: Prove that a function f : JR---+ JR
is continuous everywhere on JR iff for every open set U, f-^1 (U) is open.
That is, a function f : JR---+ JR is continuous (everywhere) iff the inverse
image of every open set is open. - Suppose f : A---+ JR is continuous on A and f(x) = 0 for all x in a dense
subset of A. Prove that f(x) = 0 for all x in A. [Hint: use sequences.]
For example, if f: A---+ JR is continuous on A and f(x) = 0 for all rational
numbers in A, then f(x) = 0 everywhere on A. - Suppose f and g are continuous on a set A and f(x) = g(x) for all x in
a dense subset of A. Prove that f(x) = g(x) for all x in A.
For example, if f and g are continuous on a set A and f(x) = g(x) for all
rational numbers in A , then f(x) = g(x) everywhere on A. - Revise the proof given in Example 5.1.12 to prove that Thomae's func-
tion has limit 0 at every real number. - Find a function f and sets A, B ~ JR such that f : A ---+ JR is continuous
and f : B ---+ JR is continuous, but f : A U B ---+ JR is not continuous. - (Project) "Additive" Functions: A function f: JR---+ JR is said to be
additive if Vx, y E JR, f(x + y) = f(x) + f(y). Suppose f is additive.
(a) Prove that Vn EN, Vx E JR, f(nx) = nf(x).
(b) Prove that Vn E Z , Vx E JR, f(nx) = nf(x).
(c) Prove that Vr E Q, Vx E JR, f(rx) = rf(x).
( d) Prove that if an additive function f : JR ---+ JR is continuous at one
point xo E JR, then it must be continuous at every x E JR.
(e) Prove that if f is continuous on JR, then Ve E JR, Vx E JR, f(cx) =
cf(x). [This means that any continuous, additive function must be
"linear" in the sense in which that word is used in a linear algebra
course.]
(f) Caution: An additive function need not be continuous, but a non-
continuous additive function must be wildly pathological. The con-
struction of such a function is beyond the scope of this book, but
can be found in Boas [16]. The graph of such a function must be
"dense" in the plane.- (Project) Suppose f : JR ---+JR is a function such that Vx, y E JR, f(x +
y) = f ( x) f (y), and that f is not the "zero" function. Prove that
(a) f(O) = 1 but there is no x such that f(x) = O;
(b) Vx, y E JR, f(x -y) = f(x)/ f(y);