 Local uniform Cauchy control for pointwise convergent continuous functions

Suppose continuous functions $f_n:\mathbb{R}\to \mathbb{R}$ converge pointwise to $f$. For each $ϵ>0$, some nonempty open interval $I$ and integer $N$ satisfy $|f_n(x)-f(x)|\le ϵ$ for every $x\in I$ and $n\ge N$. Indeed the closed sets $Q_N={\bigcap }_{n,m\ge N}\{|f_n-f_m|\le ϵ\}$ cover $\mathbb{R}$. The Baire category theorem gives an interval inside one $Q_N$, and passing $m\to \infty$ proves the assertion. The interval may depend on $ϵ$; this does not assert uniform convergence on one fixed interval.