6.8. EXERCISES 113

2. Let x→ dist(x,S) be defined in Proposition 6.0.7. Show it is uniformly continuouson Fp.

3. If ∥xn− yn∥ → 0 and xn → z, show that yn → z also. This was used in the proof ofTheorem 6.7.2.

4. 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.

5. 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.

6. Let f : D→ R be a function. This function is said to be lower semicontinuous3 atx ∈ D if for any sequence {xn} ⊆ D which converges to x it follows

f (x)≤ lim infn→∞

f (xn) .

Suppose D is sequentially compact and f is lower semicontinuous at every point ofD. Show that then f achieves its minimum on D.

7. Let f : D→ R be a function. This function is said to be upper semicontinuous atx ∈ D if for any sequence {xn} ⊆ D which converges to x it follows

f (x)≥ lim supn→∞

f (xn) .

Suppose D is sequentially compact and f is upper semicontinuous at every point ofD. Show that then f achieves its maximum on D.

8. Show that a real valued function is continuous if and only if it is both upper andlower semicontinuous.

9. Give an example of a lower semicontinuous function which is not continuous and anexample of an upper semicontinuous function which is not continuous.

10. Suppose { fα : α ∈ Λ} is a collection of continuous functions. Let

F (x)≡ inf{ fα (x) : α ∈ Λ}

Show F is an upper semicontinuous function. Next let

G(x)≡ sup{ fα (x) : α ∈ Λ}

Show G is a lower semicontinuous function.

11. Let f be a function. epi( f ) is defined as

{(x,y) : y≥ f (x)} .

It is called the epigraph of f . We say epi( f ) is closed if whenever (xn,yn) ∈ epi( f )and xn→ x and yn→ y, it follows (x,y) ∈ epi( f ) . Show f is lower semicontinuousif and only if epi( f ) is closed. What would be the corresponding result equivalent toupper semicontinuous?

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.