Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, and $\mathcal{F}$ a family of functions from $X$ to $Y$. The family $\mathcal{F}$ is equicontinuous at a point $x_0\in X$ if for every $\varepsilon > 0 $, there exists a $\delta > 0 $ such that $d_Y(ƒ(x_0),f(x) ) < \varepsilon$ for all $ƒ \in \mathcal{F}$ whenever $d_X (x_0, x) <\delta$.
Questions tagged [equicontinuity]
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Pointwise equicontinuity
Is $\Phi:=\{(t \mapsto t^n):n \in \mathbb{N}\}\subseteq C([0,1),\mathbb{R})$ where $C([0,1),\mathbb{R})$ describes the set of all continuous functions from $[0,1) \to \mathbb{R}$ pointwise equicontinuous?
The family $\Phi$ is equicontinuous at a…
user597304
- 167
0
votes
1 answer
Check a family of functions is equicontinuous
Let $g_n: [0,1]\to \mathbb{R}$ for all $n\in\mathbb{N}$ where
\begin{align*}
g_n(x)=
\begin{cases}
n^2x,\quad 0\leq x\leq \frac{1}{n}\\
\frac{1}{x},\quad \frac{1}{n}< x\leq 1.
\end{cases}
\end{align*}
I have proved that $\{g_n:n\in…
user136524
- 415
0
votes
1 answer
Equicontinuous Functions and Non-continuous functions uniformly bounded functions
I'm very confused on the idea of sequence of functions, I feel like it's very trivial and I'm overcomplicating it.
For part a, I was thinking of constructing a family of functions ${f_n}$ such that each n would have a different limit and then the…
Tia
- 25