# La crise cardiaque

## La crise cardiaque

Salut,

A-1/ A sequence of real functions $f_n \in C^2(\mathbb R)$ with $\exists h \in C(\mathbb R), \forall n\in \mathbb N, f_n '' \leq h$, and the sequence simply converge to $g$.
Is-it true that $g$ is continuous ?

A-2/ A sequence of real functions $f_n \in C^1(\mathbb R)$ with $\exists h \in C(\mathbb R), \forall n\in \mathbb N, f_n ' \leq h$, and the sequence simply converge to $g$.
Is-it true that $g$ is continuous in $\mathbb{R}-A$ with $\text{card}(A) \leq \text{card}(\mathbb N)$ ?

A-3/ A sequence of real functions $f_n \in C^3(\mathbb R)$ with :

1) $\exists h \in C(\mathbb R), \forall n\in \mathbb N, f_n ''' \leq h$,
2) $\exists N>0, \forall n \in \mathbb N, \max(|f_n'(0)|,|f_n''(0)|)\leq N$
3) the sequence simply converge to $g$.

Is-it true that $g \in C^1(\mathbb R)$ ?

Cordialement.

Dattier