Define $f(x)$ to be the least real eigenvalues of $A(x)$, then $f$ is defined in a neighborhood $B(x_0,\epsilon)$ of $x_0$ and $f(x)>1/2$ for any $x\in B(x_0,\epsilon)$. Suppose $M>\sup_x||A(x)||$, then choose $\chi(x)$ to be a nonnegative function such that $\chi$ is smooth, $\chi(x)=0$ for any $x\in B(x_0,\epsilon/2)$, and $\chi(x)=M$ for any $x\notin B(x_0,\epsilon)$. Put $B(x)=\chi(x)I$, then $B$ satisfies your requirement, since all real eigenvalues of $A+B$ is always positive, hence $A+B$ is always invertible.
Define $f(x)$ to be the least real eigenvalues of $A(x)$, then $f$ is defined in a neighborhood $B(x_0,\epsilon)$ of $x_0$ and $f(x)>1/2$ for any $x\in B(x_0,\epsilon)$. Suppose $M>\sup_x||A(x)||$, then choose $\chi(x)$ to be a nonnegative function such that $\chi$ is smooth, $\chi(x)=0$ for any $x\in B(x_0,\epsilon/2)$, and $\chi(x)=M$ for any $x\notin B(x_0,\epsilon)$. Put $B(x)=\chi(x)I$, then $B$ satisfies your requirement.