Suppose you know the following about [itex]g: \mathbb R \to \mathbb R[/itex]:(adsbygoogle = window.adsbygoogle || []).push({});

- [itex]g\in C^1[/itex];
- [itex]g\in C^2[/itex], expect at finitely many points [itex]\{x_1,\ldots,x_n\}[/itex], and [itex]|g''(x)| \leq M[/itex] (except at those points).

How can you show that there exists a sequence [itex]f_k[/itex] with the following properties?

- [itex]f_k \to g[/itex] uniformly;
- [itex]f'_k \to g'[/itex] uniformly;
- [itex]f_k \in C^2[/itex], [itex]|f''_k(x)| \leq M[/itex], and [itex]f''_k \to g''[/itex] outside [itex]\{x_1,\ldots,x_n\}[/itex].

Arzela-Ascoli is pretty much the only theorem I know of that talks about sequences of functions with uniformly bounded derivatives, but I can't for the life of me see how you could use it to construct the sequence [itex]f_k[/itex].

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# Hmmm Arzela-Ascoli, maybe?

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