WA 4
April 22, 2009
Submit this homework on Tuesday, April 28.
1. (2pts.) Let the function fon [0,1] be defined as
f(x) = 1
qif x=p
qwhere pand qare coprime integers,0≤p≤q,
0 if xis irrational or x=0.
(a) Prove that fis continuous at every irrational point and at 0, and that fis
discontinuous at every rational point of the interval (0,1];
(b) Is fRiemann integrable? If so, find R
1
R
0
f(x)dx.
(c) Is fLebesgue integrable? If so, find R
[0,1]
f.
2. (1 pt.) Show that if fis measurable real-valued function and gis a continuous
function defined on (−∞,∞) then the composition g◦fis measurable.
3. (3 pts. An extended real-valued function fis called lower semicontinuous
at the point yif f(y)6=−∞ and f(y)≤lim
x→yf(x). Similarly, fis called upper
semicontinuous at the point yif f(y)6= +∞and f(y)≥lim
x→yf(x). We say that f
is lower (upper) semicontinuous on an interval if it is lower (upper) semicontinuous
at each point of the interval. Clearly, fis upper semicontinuos if and only if −fis
lower semicontinuous.
(a) Let f(y) be finite. Prove that fis upper semicontinuous at yif and only if
given > 0, there is a δ > 0 such that f(y)≥f(x)−for all xwith |x−y|< δ.
(b) A function fis continuous (at a point of in an interval) if and only if it is
both upper and lower semicontinuous (at a point of in an interval).
(c) A step function φon [a, b] is upper semicontinuous if and only if for a partition
a=x0< x1<· · · < xn=bdefining φand for each i= 0, . . . , n the value φ(xi)
is greater than or equal to the maximal of the two values assumed on (xi−1, xi)
and (xi, xi+1) (for the endpoints x0=aand xn=b, one of the two subintervals is
void).
(d) A function fon [a, b] is upper semicontinuous if and only if there is a mono-
tone decreasing sequence {φn}of upper semicontinuous step functions on [a, b] such
that for each x∈[a, b], f(x) = lim
nφn(x).
4. (2 pts.) Let fbe a real-valued function on [a, b]. Define the lower envelope
gof fas
g(y) = sup
δ>0
inf
|x−y|<δ f(x),
and the upper envelope hof fas
h(y) = inf
δ>0sup
|x−y|<δ
f(x).
(a) For each x∈[a, b], g(x)≤f(x)≤h(x), and h(x) = f(x) if and only if fis
upper semicontinuous at x, while g(x) = h(x) if and only if fis continuous at x.
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