T/F: pointwise equicontinuous $\Leftrightarrow$ all $f_n$ continuous, uniform equicontinuous $\Leftrightarrow$ all $f_n$ uniformly continuous

Question

Is it true that:

Given a sequence of functions $(f_n)$ on a set not necessarily compact

1. pointwise equicontinuity $\Leftrightarrow$ all $f_n$ are continuous and,
2. uniform equicontinuity $\Leftrightarrow$ all $f_n$ are uniformly continuous

I read this answer (Does equicontinuity imply uniform continuity?) but it was unclear because people by and large failed to distinguish pointwise and uniform equicontinuity, resulting in ambiguous answers

Please can someone conclusively testify to the veracity of the statements!

Answer 1

I'd say 1 is definitely false. Take $f_n(x)=x^n$, on $[0,2]$. They're all contintinuous, but not equicontinuous (think about what happens around 1).

Also 2 is false. Take $f_n(x) = e^{-nx^2}$. All are uniformly continuous on $\mathbb{R}$ but for any given $\varepsilon$, the corresponding uniform $\delta$ becomes smaller and smaller as $n\to \infty$.