Real Analysis
Final Exam
Name________________________
I. By definition, the Cantor Ternary Set is the set of “left-overs” after sequentially removing all middle thirds
from the unit interval I. NOTE: This doesn’t mean that interval end points are the ONLY points left after all
middle thirds are taken out.
a. Compute the Lebesgue measure of the complement of the Cantor Set (the union of the middle thirds).
USE ONLY THE SPACE BELOW!
b. Compute the measure of the Cantor Set. USE ONLY THE SPACE BELOW!
c. Suppose a function is built as follows: Each point in the first middle third of I is mapped to 1; both
second middle thirds are constantly mapped to 1/2; the next four middle thirds are mapped to ¼, and so
on. Find the integral of the implied function on the interval [0,1]. USE ONLY THE SPACE BELOW!
II. For the following problem, consider a sample space =
power set of .
. Also suppose the Event Space is a subset of the
a. Find a probability measure on the sample space with non-zero measure on any outcome (singleton
event). USE ONLY THE SPACE BELOW!
b. Given your answer in part 1), how can you guarantee that the probability of any event in the Event
Space is between 0 and 1? USE ONLY THE SPACE BELOW!
1 t  0
III. Suppose f // (t )  
and f / (0)  1 and f (0)  2.
 1 t 0
a) Find a solution f to this differential equation that is once differentiable for each t in the real numbers.
USE ONLY THE SPACE BELOW!
b) Sketch a graph of your solution f below; draw the tangent line at the point (0, f(0)) and find its
equation. USE ONLY THE SPACE BELOW!
c) If you were riding on a particle moving on the x-axis, whose position in meters is f(t) from time t = -5
seconds to t= +5 seconds, describe the “moving experience”, in terms of where, when, and what you
would feel. USE ONLY THE SPACE BELOW!
IV. For a function to be guaranteed Riemann integrable, it must be continuous on a closed interval (or,
theoretically as good, continuous on a series of closed intervals). Continuity on a closed interval implies
uniform continuity. However, some functions that are not uniformly continuous still converge, as this exercise
shows.
a. The function f(t) = 1t is not uniformly continuous on (0,1]. Demonstrate this with care, showing exactly how
the definition of uniform continuity is violated. USE ONLY THE SPACE BELOW.
b. Integration of functions that are NOT uniformly continuous is called improper integration. Some improper
integrals converge; some do not. USING THE DEFINITION OF LIMIT, prove or disprove that the following
integral converges:

1
1
0 2 t
have to demonstrate this!
dt  lim 
x 0
1
1
x 2 t
dt . Note: The integrand is NOT uniformly continuous, but you don’t