- 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 8.4.2 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

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