- A function f is Lipschitz continuous or just Lipschitz for short if there exists a constant,
K such that
for all x,y ∈ D. Show every Lipschitz function is uniformly continuous.

- If → 0 and x
_{n}→ z, show that y_{n}→ z also. This was used in the proof of Theorem 6.7.2. - Consider f : → ℝ given by f=. Show f is uniformly continuous even though the set on which f is defined is not sequentially compact.
- 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.
- Suppose f is a function defined on D and λ ≡ inf . A sequenceof points of D is called a minimizing sequence if lim
_{n→∞}f= λ. A maximizing sequence is defined analogously. Show that minimizing sequences and maximizing sequences always exist. Now let K be a sequentially compact set and f : K → ℝ. Show that f achieves both its maximum and its minimum on K by considering directly minimizing and maximizing sequences. Hint: Let M ≡ sup. Argue there exists a sequence,⊆ K such that f→ M. Now use sequential compactness to get a subsequence,such that lim_{k→∞}x_{nk}= x ∈ K and use the continuity of f to verify that f= M. Incidentally, this shows f is bounded on K as well. A similar argument works to give the part about achieving the minimum. - Let f : D → ℝ be a function. This function is said to be lower
semicontinuous
^{3}at x ∈ D if for any sequence⊆ D which converges to x it followsSuppose D is sequentially compact and f is lower semicontinuous at every point of D. Show that then f achieves its minimum on D.

- Let f : D → ℝ be a function. This function is said to be upper semicontinuous at x ∈ D
if for any sequence ⊆ D which converges to x it follows
Suppose D is sequentially compact and f is upper semicontinuous at every point of D. Show that then f achieves its maximum on D.

- Show that a real valued function is continuous if and only if it is both upper and lower semicontinuous.
- Give an example of a lower semicontinuous function which is not continuous and an example of an upper semicontinuous function which is not continuous.
- Suppose is a collection of continuous functions. Let
Show F is an upper semicontinuous function. Next let

Show G is a lower semicontinuous function.

- Let f be a function. epiis defined as
It is called the epigraph of f. We say epi

is closed if whenever∈ epiand x_{n}→ x and y_{n}→ y, it follows∈ epi. Show f is lower semicontinuous if and only if epiis closed. What would be the corresponding result equivalent to upper semicontinuous?

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