- Let f=and let g=. Find f ⋅ g.
- Let f,g be given in the previous problem. Find f × g.
- Let f=,g=, and h=. Find the time rate of change of the box product of the vectors f,g, and h.
- Let f=. Show f is continuous at every point t.
- Suppose ≤ Kwhere K is a constant. Show that f is everywhere continuous. Functions satisfying such an inequality are called Lipschitz functions.
- Suppose ≤ K
^{α}where K is a constant and α ∈. Show that f is everywhere continuous. Functions like this are called Holder continuous. - Suppose f : ℝ
^{3}→ ℝ is given by f= 3 x_{1}x_{2}+ 2x_{3}^{2}. Use Theorem 15.7.1 to verify that f is continuous. Hint: You should first verify that the function π_{k}: ℝ^{3}→ ℝ given by π_{k}= x_{k}is a continuous function. - Show that if f : ℝ
^{q}→ ℝ is a polynomial then it is continuous. - State and prove a theorem which involves quotients of functions encountered in the previous problem.
- Let
Find lim

_{}→fif it exists. If it does not exist, tell why it does not exist. Hint: Consider along the line y = x and along the line y = 0. - Find the following limits if possible
- Suppose lim
_{x→0}f= 0 = lim_{y→0}f. Does it follow thatProve or give counter example.

- f : D ⊆ ℝ
^{p}→ ℝ^{q}is Lipschitz continuous or just Lipschitz for short if there exists a constant K such thatfor all x,y ∈ D. Show every Lipschitz function is uniformly continuous which means that given ε > 0 there exists δ > 0 independent of x such that if

< δ, then< ε. - If f is uniformly continuous, does it follow that is also uniformly continuous? Ifis uniformly continuous does it follow that f is uniformly continuous? Answer the same questions with “uniformly continuous” replaced with “continuous”. Explain why.
- Let f be defined on the positive integers. Thus D= ℕ. Show that f is automatically continuous at every point of D. Is it also uniformly continuous? What does this mean about the concept of continuous functions being those which can be graphed without taking the pencil off the paper?
- Let
Show lim

_{t→0}f= 1 for any choice of. Using Problem 11c, what does this tell you about limits existing just because the limit along any line exists. - Let f= x
^{2}y + sin. Does f achieve a maximum on the setExplain why.

- Suppose x is defined to be a limit point of a set A if and only if for all r > 0,
Bcontains a point of A different than x. Show this is equivalent to the above definition of limit point.
- Give an example of an infinite set of points in ℝ
^{3}which has no limit points. Show that if Dequals this set, then f is continuous. Show that more generally, if f is any function for which Dhas no limit points, then f is continuous. - Let
_{k=1}^{n}be any finite set of points in ℝ^{p}. Show this set has no limit points. - Suppose S is any set of points such that every pair of points is at least as far apart as 1. Show S has no limit points.
- Find lim
_{x→0}and prove your answer from the definition of limit. - Suppose g is a continuous vector valued function of one variable defined on [0,∞).
Prove
- Let U = . Determine whether U is open, closed or neither.
- Let U = . Determine whether U is open, closed or neither.
- Let U = . Determine whether U is open, closed or neither.
- Let U = . Determine whether U is open, closed or neither.
- Show carefully that ℝ
^{n}is both open and closed. - Show that every open set in ℝ
^{n}is the union of open balls contained in it. - Show the intersection of any two open sets is an open set.
- If S is a nonempty subset of ℝ
^{p}, a point x is said to be a limit point of S if Bcontains infinitely many points of S for each r > 0. Show this is equivalent to saying that Bcontains a point of S different than x for each r > 0. - Closed sets were defined to be those sets which are complements of open sets. Show that a set is closed if and only if it contains all its limit points.
- Prove the extreme value theorem, a continuous function achieves its maximum and
minimum on any closed and bounded set C. Hint: Suppose λ = sup. Then there exists⊆ C such that f→ λ. Now select a convergent subsequence. Do the same for the minimum.
- Let C be a closed and bounded set and suppose f : C → ℝ
^{m}is continuous. Show that f must also be uniformly continuous. This means: For every ε > 0 there exists δ > 0 such that whenever x,y ∈ C and< δ, it follows< ε. It is in the chapter but go over it again. This is a good time to review the definition of continuity so you will see the difference. Hint: Suppose it is not so. Then there exists ε > 0 andandsuch that<but≥ ε. - A set whose elements are open sets C is called an open cover of H if ∪C⊇ H. In
other words, C is an open cover of H if every point of H is in at least one
set of C. Show that if C is an open cover of a closed and bounded set H
then there exists δ > 0 such that whenever x ∈ H, Bis contained in some set of C. This number δ is called a Lebesgue number. Hint: If there is no Lebesgue number for H, let H ⊆ I = ∏
_{i=1}^{n}. Use the process of chopping the intervals in half to get a sequence of nested intervals, I_{k}contained in I where diam≤ 2^{−k}diamand there is no Lebesgue number for the open cover on H_{k}≡ H ∩ I_{k}. Now use the nested interval theorem to get c in all these H_{k}. For some r > 0 it follows Bis contained in some open set of U. But for large k, it must be that H_{k}⊆ Bwhich contradicts the construction. You fill in the details. - Here is another way to consider this Lebesgue number. If there is no Lebesgue
number, then for each n ∈ ℕ, 1∕n is not a Lebesgue number. Hence there exists
x
_{n}∈ H such that Bis not contained in a single set of C. Extract a convergent subsequence, still denoted as x_{n}→ x. Then Bis contained in a single set of C. Isn’t it the case that Bis contained in Bfor all n large enough? Isn’t this a contradiction? - A set is compact if for every open cover of the set, there exists a finite
subset of the open cover which also covers the set. Show every closed
and bounded set K in ℝ
^{p}is compact. Next show that if a set in ℝ^{p}is compact, then it must be closed and bounded. This is called the Heine Borel theorem. Hint: To show closed and bounded is compact, you might use the technique of chopping into small pieces of the above Problem 35. You could also do something like the following. Let δ be a Lebesgue number for the open cover C of K. Now consider B. If it covers K you are done. Otherwise, pick x_{2}not in it. Consider B. If these two balls cover K, then you are done. Otherwise pick x_{3}not covered. Continue this way. Argue the the sequential compactness of K requires this process to stop in finitely many steps. If a set K is compact, then it obviously must be bounded. Otherwise, you could consider the open cover_{n=1}^{∞}. It the set K is not closed, then there is a point not in K called x and a sequence of pointsof K converging to x. Explain why F_{m}≡∪_{k=m}^{∞}x_{k}is closed. Consider the increasing sequence of open sets F_{m}^{C}. - Suppose S is a nonempty set in ℝ
^{p}. DefineShow that

Hint: Suppose dist

< dist. If these are equal there is nothing to show. Explain why there exists z ∈ S such that< dist+ ε. Now explain whyNow use the triangle inequality and observe that ε is arbitrary.

- Suppose H is a closed set and H ⊆ U ⊆ ℝ
^{p}, an open set. Show there exists a continuous function defined on ℝ^{p}, f such that f⊆, f= 0 if xU and f= 1 if x ∈ H. Hint: Try something likewhere U

^{C}≡ ℝ^{p}∖U, a closed set. You need to explain why the denominator is never equal to zero. The rest is supplied by Problem 38. This is a special case of a major theorem called Urysohn’s lemma.

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