Are strictly concave functions robust to "small changes"?

Question

Let $f:\mathbb{R}^n\times \mathbb{R}^m\rightarrow \mathbb{R}$ be a continuous function. Suppose that there exists an $a_0\in\mathbb{R}^m$ such that $f\left(x,a_0\right)$ is strictly concave in $x$. Is it true that there exists a ball $A$ around $a_0$ such that $f\left(x,a\right)$ is concave in $x$ for $a\in A$?

Answer 1

This not true; essentially all we need to do is set $$f(x,a)=-x^2+ag(x)$$ and pick $g(x)$ to be any function whose second derivative is not bounded above. Then $$\frac{d}{dx}\frac{d}{dx}f(x,a)=-2+ag''(x).$$ Since $g''(x)$ is not bounded above, there is some $x$ for which this is positive for any $a>0$.

An explicit example of this would be $$f(x,a)=-x^2+a\sin(x^2).$$

You can even do a little worse - consider a function like $f(x,a)=-x^2+a|x|$. This gives a counterexample even you wanted $f$ to be a function $f:[-1,1]\times \mathbb R\rightarrow \mathbb R$.