This question already has an answer here: Does pointwise equicontinuous and uniformly equicontinuous implies compactness? 1 answer

Can anyone think of an example of such set of functions?(If domain is compact then pointwsie equicontinuity implies uniformly equicontinuous)

Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$. If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded?

I’m stuck in an analysis problem, it states: Let $\lbrace f_n:[0,1)\rightarrow \mathbb{R}\rbrace_{n\in \mathbb{N}}$, with $f_n(x)=x^n$. Prove that: $i)$ The familiy is pointwise equicontinuous (i.e. for every $x_0\in[0,1)$. $ii)$ The familiy is NOT uniformly equicontinuous. My attempt: Por $i)$, we need to prove that for every $x_0\in[0,1)$, and $\forall \varepsilon>0 \quad \exists \delta=\delta(\varepsilon;x_0)>0$ such that if […]

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