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Given a sequence of continuous functions $f_n: [a,b] \rightarrow \mathbb{R}$, $f_n \rightarrow f$ pointwise and $f$ continuous, does $f_n \rightarrow f$ uniformly?

I know that this is not true if $f$ is not continuous, for example let $[a,b] = [0,1]$ and $f_n = x^n$, then $f_n \rightarrow f$, $f(x) = 0$ for $x \in [0,1)$ and $f(1) = 1$.

However, given $f_n \rightarrow f$ and $f$ is continuous, this seems to be true. It seems really strong though so I am wondering if there are counterexamples.

CuriousAlpaca
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1 Answers1

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A standard counterexample is $f_n(x)=nxe^{-nx}$ in $[0,1]$. It converges pointwise to the zero function. However, the convergence is not uniform, because for every $n\in\mathbb{N}$:

$\sup_{x\in [0,1]}|f_n(x)-0|\geq |f_n(\frac{1}{n})|=\frac{1}{e}$

I'll add that a sufficient condition for uniform convergence is Dini's theorem. If we remove any of its requirements (like here we removed the monotonicity of the sequence at every point) then there are counterexamples.

Mark
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  • I know this was posted like 3 years ago, but just wanted to ask, actually for dini's theroem ,is it possible to loosen the monotone condition to allow for a finite N before which the sequence is not monotone decreasing and after that it is monotone decreasing? – Burger crazy Oct 02 '23 at 14:01
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    @Burgercrazy It definitely makes no difference if that is the same $N$ for all points $x$. However, if $N$ depends on $x$, this is not true anymore. See here: https://math.stackexchange.com/questions/1553143/dinis-theorem-and-monotonicity – Mark Oct 02 '23 at 15:57