A question on the assumptions in the theorem for changing limit and derivative for sequence of functions

Question

I am looking for an example for a sequence of differentiable functions $\{f_n\}$ on a closed bounded interval $[a,b]$ such that $\{f_n\}$ converges uniformly to a differentiable function $f$ , $\{f_n'\}$ converges point-wise to a function $g$ on $[a,b]$ such that $f'=g$ but $\{f_n'\}$ does not converge uniformly on $[a,b]$ . Please help . Thanks in advance

Answer 1

We take $[a,b] = [0,1]$, and we build $\{f_n\}$ which converges uniformly to $0$ (while $\{f'_n\}$ converges pointwise to $0$).

Let us define the derivative (for $n \geqslant 2$) :

• $\forall x \in [0,\frac{1}{n}]$, $f'_n(x) = nx$

• $\forall x \in [\frac{1}{n}, \frac{2}{n}]$, $f'_n(x) = 2 - nx$

• $\forall x \in [\frac{2}{n}, 1]$, $f'_n(x) = 0$

The graph of the derivative is :

Computing the area of the triangle, you have immediately $0 \leqslant f_n \leqslant \frac{1}{n}$

Hence $\{ f_n \}$ converges uniformly to $0$. And $f'_n$ also converges pointwise to $0$, but not uniformly : $\sup \limits_{x \in [0,1]} f'_n(x) = 1$, for all $n$.