Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ such that:

- $f_n$ is strictly convex on $(-\infty,x_n)$,
- $f_n$ is strictly concave on $(x_n, +\infty)$.

Finally, let assume that the sequence uniformly converges to $f\in\mathscr{C}^2(\mathbb{R})$.

My feeling is that the sequence $x_n$ has to converge. Can one has an idea to show this ?

Thank you!