Super-level sets of Hardy–Littlewood maximal function are open?

I am working on the book Measure, Integration and Real Analysis by Sheldon Axler. I am stuck on Problem 9 of Section 4A. For $h: \mathbb{R} \to \mathbb{R}$ Lebesgue measurable, $h^*$ is defined as follows:

$$ h^*(x)=\sup_{r>0}\frac{1}{|B(x,r)|} \int_{B(x,r)} |h(y)|dy.$$

Suppose $h: \mathbb{R} \to \mathbb{R}$ is Lebesgue measurable. Prove that$$\left\{ b \in \mathbb{R} : h^{\ast}(b) > c\right\}$$is an open subset of $\mathbb{R}$ for every $c \in \mathbb{R}$.

My Idea is like this: Let $A=\left\{ b \in \mathbb{R} : h^{\ast}(b) > c\right\}$. Pick $b \in A.$ I want to show that there is a ball $B(b,\epsilon)$ contained inside of $A$. But I don't know how to get to this.Any help?