Given closed interval $[a,b]$. How to show that if we have a sequence of continuous functions $\{f_n\}$ point wise converging to $f$, then $f$ does not have infinite discontinuities?
By infinite discontinuity, I mean the one-sided limits don’t go to $+\infty$ or $-\infty$
Let $f_n$ being equal to $\frac 1 x$ over $(\frac 1 n,1]$ and joining linearly $0$ to $n$ between $x=0$ and $x=\frac 1 n$. In particular, $f_n$ are continuous.
That function sequence converges pointwise to $f$ being $\frac 1 x$ on $(0,1]$ and $0$ at $x=0$, which is not bounded.