4.11. EXERCISES 113

4.11 Exercises1. A function f is Lipschitz continuous or just Lipschitz for short if there exists a con-

stant, K such that| f (x)− f (y)| ≤ K |x− y|

for all x,y ∈ D. Show every Lipschitz function is uniformly continuous.

2. If |xn − yn| → 0 and xn → z, show that yn → z also. This was used in the proof ofTheorem 4.7.2.

3. Consider f : (1,∞) → R given by f (x) = 1x . Show f is uniformly continuous even

though the set on which f is defined is not sequentially compact.

4. If f is uniformly continuous, does it follow that | f | is also uniformly continuous? If| f | is uniformly continuous does it follow that f is uniformly continuous? Answer thesame questions with “uniformly continuous” replaced with “continuous”. Explainwhy.

5. Suppose f is a continuous function defined on D and λ ≡ inf{ f (x) : x ∈ D} . A se-quence {xn} of points of D is called a minimizing sequence if

limn→∞

f (xn) = λ .

A maximizing sequence is defined analogously. Show that minimizing sequencesand maximizing sequences always exist. Now let K be a sequentially compact setand f : K → R. Show that f achieves both its maximum and its minimum on Kby considering directly minimizing and maximizing sequences. Hint: Let M ≡sup{ f (x) : x ∈ K} . Argue there exists a sequence, {xn} ⊆ K such that f (xn)→ M.Now use sequential compactness to get a subsequence,

{xnk

}such that limk→∞ xnk =

x ∈ K and use the continuity of f to verify that f (x) = M. Incidentally, this shows fis bounded on K as well. A similar argument works to give the part about achievingthe minimum.

6. Let f : D → R be a function. This function is said to be lower semicontinuous3

•

x

Lower semicontinuous at x

at x ∈ D if for any sequence {xn} ⊆ D which converges to x it follows f (x) ≤liminfn→∞ f (xn) . Suppose D is sequentially compact and f is lower semicontinu-ous at every point of D. Show that then f achieves its minimum on D.

7. Let f : D → R be a function. This function is said to be upper semicontinuousat x ∈ D if for any sequence {xn} ⊆ D which converges to x it follows f (x) ≥limsupn→∞ f (xn) . Suppose D is sequentially compact and f is upper semicontin-uous at every point of D. Show that then f achieves its maximum on D.

3The notion of lower semicontinuity is very important for functions which are defined on infinite dimensionalsets. In more general settings, one formulates the concept differently.