Math 3210-1
Example from Class
Recall Theorem 101 says: Suppose that f is integrable on [a, b] and g is continuous on [c, d], where
f ([a, b]) ⊆ [c, d]. Then g ◦ f is integrable on [a, b].
We tried to think of an example where f and g are integrable, but the composition is not. Here is such
an example:
Let f be the Dirichlet function given by f : [0, 1] → [0, 1] with
1
if x = m
n
n where m and n are in lowest terms
f (x) =
0
if x ∈
/Q
Let g : [0, 1] → [0, 1] be given by
g(x) =
1
0
if x 6= 0
if x = 0
We proved in class that f is integrable on [0, 1]. We will now prove that g is integrable on [0, 1].
Let P = {x0 , x1 , . . . , xn } be any partition of [0, 1]. Then Mi = mi = 1 for all i = 1, . . . n, so U (g, P ) =
n
n
X
X
∆xi = 1−0 = 1. Similarly we have L(g, P ) = 1 for any partition P . Thus U (g) = 1 = L(g),
Mi ∆xi =
i=0
i=0
so g is integrable.
Finally notice that h(x) = (g ◦ f )(x) =
integrable.
1
0
if x ∈ Q
, and we proved in class that this function is not
if x ∈
/Q