Give an example of a sequence of functions having the following property.

Question

I am finding an example of a sequence $(f_n)$ of differentiable functions on an interval $I$ such that $f_n\to 0$ uniformly on $I$ but $f'_n(x)\to \infty$ for all $x\in I$ as $n\to \infty$.Can someone provide me an example where this can occur.I was trying with trigonometric functions like $f_n(x)=\sin(nx)/n$ etc but they are not working.

Answer 1

Partial answer: such an example has to be weird. Assuming continuity of the derivatives I will show that no such sequence can exist.

Write $I$ as $\cup_m\{x:f_n'(x) \geq 1 \forall n >m\}$. By Baire Catrgory Theorem there is an integer $m$ and interval $[a,b] \subset I$ (with $a <b$) such that $f_n'(t) >1$ for all $t \in [a,b]$ for all $n \geq m$. But now we get a contradiction from $f_n(b)-f_n(a)=\int_a^{b} f_n'(t)dt \geq b-a$ for all $n >m$.