Analysis III TT23, Differentiation and limits

Suppose that $f _ n : [a, b] \to \mathbb R$ is a sequence of functions such that

• $f _ n$ is continuously differentiable on $(a, b)$
• $f _ n$ converges pointwise to some function $f$ on $[a, b]$.
• $f’ _ n$ converges uniformly to some bounded function $g$ on $[a, b]$

then $f$ is differentiable and

\[\lim _ {n \to \infty} f'\ _ n = \left(\lim\ _ {n \to \infty} f\ _ n\right)'\]

Suppose that $\phi _ i : [a, b] \to \mathbb R$ is a sequence of functions continuously differentiable on $(a, b)$ with $\sum _ i \phi _ i$ converging pointwise. Suppose that $ \vert \phi _ i’(x) \vert \le M _ i$ for all $x \in (a, b)$ where $\sum _ i M _ i < \infty$. Then $\sum \phi _ i$ is differentiable and

\[\left(\sum _ i \phi _ i\right)' = \sum _ i \phi _ i'\]

\[f _ n(x) = \frac 1 n \sin(n^2 x)\]