Math Problems Vault

Problem 7.1.1

(a)(b)

(SC4b) If $f(\boldsymbol {x})=-\infty$ , then $f$ is USC at $\boldsymbol {x}$ iff $\lim_{\boldsymbol {y} \rightarrow \boldsymbol {x}} f(\boldsymbol {y})= -\infty$ . (SC7b) $\bar{f}$ is USC, and is the minimal USC function $\geqslant f$ . (SC9) Let $A$ denote a generic nonempty index set. For each $\alpha \in A$ , suppose that $f_{\alpha }$ is a function from $\mathbb {R}^{k}$ into $\overline{\mathbb {R}}$ . We have the following:

?

(a)

If $f_{\alpha }$ is LSC for each $\alpha \in A$ , then $\sup_{\alpha \in A} f_{\alpha }$ is LSC.

(b)

If $f_{\alpha }$ is USC for each $\alpha \in A$ , then $\inf_{\alpha \in A} f_{\alpha }$ is USC.

Problem 7.1.2

(a)(b)(c)

Let $E \subset \overline{\mathbb {R}}^{k}$ .

?

(a)

$E$ is open iff $\; \mathbb {1}_{E}$ is LSC.

(b)

$E$ is closed iff $\; \mathbb {1}_{E}$ is USC.

(c)

We have $\underline{\; \mathbb {1}_{E}}=\; \mathbb {1}_{E^{\circ }}$ and $\overline{\; \mathbb {1}_{E}}=\; \mathbb {1}_{\bar{E}}$ .

Problem 7.1.3

(a)(b)

?

(a)

If $f_{1}$ and $f_{2}$ are LSC [or USC] and finite on $\mathbb {R}^{k}$ , then $f_{1}+f_{2}$ is LSC [or USC].

(b)

If $\left\{ f_{n}\right\}_{n=1}^{\infty }$ is a sequence of nonnegative LSC [or USC] functions, is $\sum_{n=1}^{\infty } f_{n}$ necessarily LSC [or USC]?

Problem 7.1.4

(a)(b)

Let $K$ denote a compact subset of $\mathbb {R}^{k}$ .

?

(a)

If $f$ is a real-valued function that is USC on $K$ , then $f$ obtains a local maximum on $K$ .

(b)

If $f$ is a real-valued function that is LSC on $K$ , then $f$ obtains a local minimum on $K$ .

Problem 7.1.5

(a)(b)(c)

Let $A$ denote a generic nonempty index set, and suppose for each $\alpha \in A$ that $f_{\alpha }$ is continuous.

?

(a)

It is not necessarily the case that $\sup_{\alpha \in A} f_{\alpha }$ is continuous.

(b)

$\sup_{\alpha \in A} f_{\alpha }$ is Borel measurable.

(c)

It is not necessarily the case that $\inf_{\alpha \in A} f_{\alpha }$ is continuous, but at any rate it is still Borel measurable.

Problem 7.1.6

(a)(b)(c)

For each $\boldsymbol {x} \in \mathbb {R}^{k}$ and $\delta >0$ , define $\phi (\boldsymbol {x})=\inf_{\delta >0} \sup_{\boldsymbol {s}, \boldsymbol {t} \in B(\boldsymbol {x}, \delta )}\left|f(\boldsymbol {s})-f(\boldsymbol {t})\right|$ .

?

(a)

$\phi$ is USC, hence Borel measurable.

(b)

$f$ is continuous at $\boldsymbol {x}$ iff $\phi (\boldsymbol {x})=0$ .

(c)

$\left\{ \boldsymbol {x} \in \mathbb {R}^{k}: f \text{ is continuous at } \boldsymbol {x}\right\}$ is a Borel set. This shows that for any $f$ , the continuity set of $f$ [as well as the set of points where $f$ is discontinuous] is a Borel set.