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I found a very elegant construction in this link: https://math.stackexchange.com/a/311289/444015, I have no questions about this example. However, at least for me, it doesnt look like a natural construction.

I tried to find more direct examples, but I didnt succeed. Every smooth equicontinuous sequence that I found is bounded.

I was wondering if there is any more obvious example than this, or if indeed to find any example, it is necessary to construct the sequence of functions in a similar way.

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Let $f_n(x)=\frac 1n\sin(x^2)$. Then $f_n\to0$ uniformly, so $(f_n)$ is equicontinuous.

(That gives $\sup_x|f_n'(x)|=\infty$. "With unbounded derivatives" is somewhat ambiguous; if you want $\sup_n|f_n'(0)|=\infty$ use $\frac 1n\sin(n^2x)$.)

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  • $\begingroup$ I got it! I believe that the question asks for $\sup_{x}$. Thank you! $\endgroup$ – Lucas Corrêa Jan 11 at 14:55

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