Supremum conditions for equicontinuity

Question

Suppose $\mathcal{E} \subset C_0([a, b], \mathbb R)$ is a family of functions.

1. Show that $g(x) = \{\sup f(x) : f \in \mathcal{E}\}$ is continuous does not imply that $\mathcal{E}$ is equicontinuous.

2. 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.

Answer 1

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