A. Suppose $f$ is a real continuous function on $\mathbb{R}$, $f_n(t)=f(nt)$ for $n=1,2,3,...,$ and $\{f_n\}$ is equicontinuous on $[0,1]$. Then $f$ is constant.
B. Therefore, if $f$ is not constant, then $\{f_n\}$ is not equicontinuous on $[0,1]$.
Since the proof A. doesn't rely on the value of 1 from the interval, B statement is also true for $[0,m]$, for any positive number $m$.
Since if A is true, then A is also true for $[-1,0]$, same for B statement, and thus B statement can be true for interval $[-m, m]$? What about $[a,b]$ $\forall a,b\in\mathbb{R}$? If not then what's the importance of the number $0$?