TITLE: Supremum conditions for equicontinuity
QUESTION [5 upvotes]: Suppose $\mathcal{E} \subset C_0([a, b], \mathbb R)$ is a family of functions.

Show that $g(x) = \{\sup f(x) : f \in \mathcal{E}\}$ is continuous does not imply that $\mathcal{E}$ is equicontinuous.
If $g(x) = \{\sup f(x) : f \in \mathcal{F}\}$ is continuous for every $\mathcal{F} \subset \mathcal{E}$, is $\mathcal{E}$ equicontinuous?

I need a hint for these two questions. I don't even know where to start.

REPLY [4 votes]: Hint: Consider
$$
f_n(x)=\left\{\begin{array}{}
0&\text{for }x\lt\frac12-\frac1{2n}\\
1-n\left(1-2x\right)&\text{for }\frac12-\frac1{2n}\le x\le\frac12\\
1&\text{for }x\gt\frac12\\
\end{array}\right.
$$