I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function , but not converges uniformly.
My question is the following :–
Let $f:[0,1]\rightarrow \Bbb R$ be continuous , does there always exists a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ such that $f_n\rightarrow f$ pointwise, but not uniformly on $[0,1]$.
Best Answer
Yes, there is always such a function.
Take a sequence of continuous functions $g_n$ which converges pointwise, but not uniformly, to $0$. Then $f_n=f + g_n$ converges pointwise, but not uniformly to $f$.
For an example of such $g_n$, you can take the following: $$ g_n(x) = \cases{ nx & if $x< \frac1n$\\ 2-nx & if $\frac1n \leq x < \frac2n$\\ 0& otherwise} $$ The graph of $g_n$ will be a triangle starting at $(0,0)$, going up to $(1, \frac1n)$, then down to $(\frac2n, 0)$, and then flat horizontal from there. This does converge pointwise to $0$ (as at any non-zero point $a\in [0,1]$, eventually $\frac2n<a$, and $g_n(a) = 0$), but not uniformly, as it always has a maximum value of $1$.