Uniform Convergence and differentiable functions I have been working on this textbook question and am not sure what to do. Is there a sequence of differentiable functions on some interval, say [0,2], converging to 0 uniformly, but where $f'_n(1)$ does not converge to 0? Thank you.

Consider the sequence of functions
$$
  f_n:x \to \frac{\sin (nx)}{n} \text{.}
$$
With increasing $n$, $f_n$ starts to "wobble" faster and fastter, but with decreasing amplitude. The latters makes $f_n$ converge to the zero function uniformly, since $f_n$ is bounded by $\frac{1}{n}$ and thus $|f_n(x)-f_n(y)| < \frac{2}{n}$ for all $x,y$. Since $$
  f_n'(x) = cos(nx) - \frac{\sin(nx)}{n}
$$
you have $f_n'(2\pi k) \to 1$, though. This caused by the "wobbling" getting faster and faster, which offsets the decreasing ampltitude, as the derivative shows (the $n$ in the denominator is cancelled by the $n$ you get from differentiating $\sin(nx)$). You can even make it diverge if instead of $\sin(nx)$ you use $\sin(n^2 x)$ in the definition of $f_n$.
