From the definition of a equicontinuous family, it's known that if $ \left |x-y \right |< \delta$ then $\left |f(x)-f(y) \right |< \varepsilon$ for all $f$ in the family.
For all functions in the family, is the delta the same?
For example, the family $f_n (x) = nx$ is equicontinuous? I think not, since the delta that's taken is $ \delta = \frac{\varepsilon}{n} $ and this changes...
Yes, it's always the same $\delta$, for every $f$.
And, yes, the family $(f_n)_{n\in\Bbb N}$ of functions from $\Bbb R$ into $\Bbb R$ defined by $f_n(x)=\frac xn$ is not uniformly continuous. Of course, the fact do your $\delta$'s are not the same for each $f_n$ doesn't prove that, but it suggests that it is true.