# Sequence of smooth equicontinuous functions with unbounded derivatives

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.

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 $$\frac 1n\sin(n^2x)$$.)
• I got it! I believe that the question asks for $\sup_{x}$. Thank you! – Lucas Corrêa Jan 11 at 14:55