Let ${f_n}$ be a sequence of functions $E\to \mathbb{R}$ and suppose $E= \cup _{k=1} \;E_k$ for some sets $E_k$

Question

Let ${f_n}$ be a sequence of functions $E\to \mathbb{R}$ and suppose $E= \cup _{k=1} \;E_k$ for some sets $E_k$. Suppose there is a function $f:E→\mathbb{R} $ such that $f_n→f$ uniformly on each $E_k$. Decide, with justification, if this implies that $f_n→f$ uniformly on $E$.

Is this statement is true..if it is not true under what conditions the statement is true..can any help me please

Answer 1

No, $(0,1]=\displaystyle\bigcup_{k=2}\left[\dfrac{k-1}{k},1\right]$ and $f_{n}(x)=\dfrac{1}{x}\chi_{[(k-1)/k,1]}(x)$ and $f(x)=\dfrac{1}{x}$.