Math 301 Homework
Donnay, Fall 2011
Assignment Wk 9 Part 1: Due Wednesday, Nov 9 at 5pro.
S1. Consider the set {x : x > 1}.
a. Prove using the boundm'y definition that this set is open.
b. Prove using the hall definition that this set is open.
$2. Consider the set {x : x < 1},
a, Prove using the boundary definition that this set is closed.
b. Prove that this set is closed by considering the complement of this set.
$3. Consider the statement: Every point of the set A is contained in the set B. What is negation
of this statement? ie. It is not the case that every point of the set A is contained in B. Use Be in
your answer.
Morgan: Ch. 5 ÿ 7 (create your example so that NUk is a closed interva! of positive length).
$4. a. Let f(x) = x2. For U = (1,4), what is f-l(U)? Illustrate your answer by drawing a
graph of y = zu.
b. Let f(x) = x/ÿ. For U = (1,4), what is f-l(U)? Illustrate your answer by drawing a graph
of y
,ÿ,
Assigmnent Wk 9 Part 2: Due Friday Nov 11 at 5pin.
S1. In class, we illustrated the meaning of the three equivalent conditions of continuity by
drawing a picture of a discontinuous function and showing that the conditions did not hold. Repeat
this activity But now using a function that is defined by formulas.
a. Define, via formulas, a function f(x) that is defined for all x € R and that is discontinuous
at x = 1. Draw a picture of this function.
b. Use the 5 ÿ condition to show that this function is not continuous at ac = 1.
c. Use the sequence definition to show that this function is not continuous at x - 1.
b. Use the open set condition to show that this function is not continuous at x = 1.
$2. a. Consider the function f(x)
x2. Prove that this function is continuous at every point
p € R by using the sequence condition for continuity.
2
b. Prove that this function is continuous using the limit laws and the fact that we have proven
g(x) = x is continous.
$3. (Ch 6) Write out the proof that condition (1) implies condition(3). Add extra explanations
so it Iaakes seiase to you. Try to illustrate the proof with a diagram (or severM diagrams).
Morgan, Ch. 7 €ÿ i (state the domain of the composition function), 2
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