limit of positive concave function is concave

Question

Let $A\subsetneq \mathbb{R}^{n}$( A is compact). Consider $f:\mathbb{R}^{n}\rightarrow \mathbb{R}_{+}^{n}\cup\{0\}$ is continuous map. Suppose that for all $a\in A$ , $\alpha \mapsto f(\alpha)$ is concave positive function.

Let $\beta_{n} \in A$. We can show that if $f(\beta_{n})\rightarrow f(\beta)$, then $f(\beta)>0$.

Can one give an counter example for above question when concavcity drop?

Thanks in advance.

Answer 1

The statement is wrong. Here is a counterexample.

Let $A=(-1,1)$, $f(x) =\max(1- |x|,0)$, $\beta_n = 1 + \frac1n$. Then $\beta_n\to 1$, $f(\beta_n)\to0$.