Differentiable Pointwise Limit Function of a Sequence of Differentiable Function.

We know that a point-wise limit function $f$ of a sequence of differentiable functions $(f_n)$ is not necessarily differentiable. The sequence of differentiable functions $$f_n(x) = \sqrt(x^2+\frac{1}{n^2})$$ has this property. Here, $f_n \rightarrow f$ poitwise where $f(x) = |x|$ which is not differentiable at $x=0$. My question is: Is it possible to find a sequence of differentiable functions with differentiable pointwise limit function? But yet, $$f_n^{\prime}(x) \nrightarrow f^{\prime}(x).$$

1 Answer

If $f_n(t)=\sin(nt)/n$ then $f_n\to0$ uniformly although $f_n'(0)=1$ for all $n$.