Recall that you may be asked to present your solutions to Bronze questions in class and that
Bronze Questions are to be turned in at the end of class.
For Wednesday, April 22 (Note the slightly extended due date.)
• Get 7–9 hours of good sleep each night. Sleep is the basis for memory and creative
thought, so your sleep time should be as regular as possible and absolutely nonnegotiable. A cold, pitch-black environment, with absolutely no blue light can help. More
sleep tips can be found online.
Note: Losing even an hour of sleep can strongly impact your immune system’s performance.
I Read Sections 6.4 and 6.5 in Cummings.
• Bronze Questions
1. (Exercise 6.5 in Cummings) Let f : R → R be given by f(x) = x
2 + 2x + 1. Use the -δ
definition of continuity to prove that f is continuous at x = 1.
2. (Exercise 6.6(a) in Cummings) Let f : R → R be given by f(x) = |x|. Use the –
δ definition of continuity to prove that f is continuous on R. [Hint: Use the reverse
triangle inequality; see page 17.]
3. (Exercise 6.13 in Cummings) Let f : R → R be given by f(x) := (
1 if x ≥ 0,
−1 if x < 0.
Use the -δ definition of continuity to prove that f is discontinuous at x = 0.
4. (Exercise 6.26 in Cummings) Let f : R → R be given by g(x) := (
, if x 6= 0,
0, if x = 0.
Prove that f is continuous at x = 0. [Drawing a picture might help. You can assume
standard calculus facts about the sine function.]
5. Consider the function g defined above in exercise 4. Compute the derivative g
x 6= 0 (using usual calculus tools such as the product rule and chain rule). Is it possible
to assign a value to g
(0) to make g
(x) continuous at x = 0?
6. A function f : R → R is called Lipschitz continuous if there exists a constant C > 0 such
|f(x) − f(y)| ≤ C|x − y| for all x, y ∈ R
Use the -δ definition of continuity to prove that f is Lipschitz continuous, then it is
Rudolf Lipschitz (1832 – 1903)
Lipschitz continuity is perhaps one of the most useful forms of continuity in all of mathematics. A classic example of a function which is continuous but not Lipschitz continuous
is f(x) = p
|x|. He was a student of Dirichlet, and a great mathematician. His only
student was Felix Klein, of Klein-bottle fame. Some people make fun of Lipschitz’s name,
but nobody can make fun of his mustache