Upper and Lower semicontinuity and right-left continuity.

Question

Let $f: [0,1]\rightarrow\mathbb{R}$ be a function. Is there any relation between upper-lower semicontinuity of $f$ and right-left continuity of $f$?

I mean, is there any implication?

Thanks in advance.

Answer 1

The only implication is that both left + right continuity and upper + lower semicontinuity are equivalent (and, of course, are equivalent to continuity). Otherwise, we can create a table of counterexamples. Here, we define several real functions of a variable $x$, which possess the given continuity properties at $0$:

$$\begin{array}{|c|c|} \hline & \text{Left-continuous} & \text{Right-continuous} & \text{Both} & \text{Neither} \\ \hline \text{USC} & \begin{cases} 1 & \text{if }x \le 0 \\ 0 & \text{if } x > 0 \end{cases} & \begin{cases} 0 & \text{if }x < 0 \\ 1 & \text{if } x \ge 0 \end{cases} & - &\begin{cases} 0 & \text{if }x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \\ \text{LSC} & \begin{cases} 0 & \text{if }x \le 0 \\ 1 & \text{if } x > 0 \end{cases} & \begin{cases} 1 & \text{if }x < 0 \\ 0 & \text{if } x \ge 0 \end{cases} & - &\begin{cases} 1 & \text{if }x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \\ \text{Both} & - & - & 0 & - \\ \text{Neither} & \begin{cases} 0 & \text{if }x \le 0 \\ \sin\left(\frac{1}{x}\right) & \text{if } x > 0 \end{cases} & \begin{cases} \sin\left(\frac{1}{x}\right) & \text{if }x < 0 \\ 0 & \text{if } x \ge 0 \end{cases} & - & \begin{cases} \sin\left(\frac{1}{x}\right) & \text{if }x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \\ \hline \end{array}$$