Right-const function and pointwise/uniform convergence

Question

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. Determine whether $f$ must be right-const if:

• a) $\{f_n\}$ is pointwise convergent to $f$
• b) $\{f_n\}$ is uniformly convergent to $f$

I had spent much time trying to find counterexample for a) or b) but I gave up. Maybe it is possible to prove a) or b)?

Answer 1

For a), how about $f_n(x) =\min(x,n)$?

For b): Let $f(x)=\sin(x)/x$ for $x>0$ and $=0$ for $x\le 0$. And $f_n(x) =f(x)$ for $x\le n\pi$ and $ =0$ for $x\ge n\pi$.