MATH 4100 Assignment #2 – Due Tuesday February 27, 2018
Problem #1
Consider the following prompts and questions used to help characterize functions
࠵?: ࠵? → ࠵?
with the
property that
࠵?
࠵? + ࠵?
= ࠵?
࠵?
+ ࠵?
࠵?
i.
What is a set of examples and non-examples?
ii.
What is
࠵?
0
for all such functions? What functions does this eliminate?
iii.
How would you describe all possible values of
࠵?(࠵?)
, for
࠵?
a rational number, given that
࠵?(1) =
࠵?
, where
࠵?
is any real number? Why are there no other possible answers?
iv.
How can this be used to clearly and simply show that
࠵?
࠵?
= ࠵?
0
is not a possible solution?
v.
How can you use the added assumption that the function is differentiable at all real numbers to
extend your answer in (iii) to all real numbers?
Determine your own prompts and questions to characterize the functions
࠵?: ࠵?
0
→ ࠵?
0
with the property
that
࠵?
(࠵?
1
, ࠵?
1
) + (࠵?
0
, ࠵?
0
)
= ࠵?
(࠵?
1
, ࠵?
1
) + ࠵?
(࠵?
0
, ࠵?
0
)
and provide your own characterization.