Consider a compact (semialgebraic) ball $B\subset \mathbb{R}^n$ and a semialgebraic set $A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n$ defined through some representation in terms of polynomials $f_1,\ldots,f_s\in \mathbb{R}_m[x_1,\ldots, x_n]$.

Is the dependence of $A(f_1,\cdots,f_s)\cap B$ on the polynomials $f_1,\cdots,f_s$ outer semicontinuous?

Are there known results like this in the theory of real semialgebraic sets?

A set-valued function $F:X\to \mathcal{P}(\mathbb{R}^n)$, that to each $x\in X$ associates a compact subset $F(x)\subset\mathbb{R}^n$, is called outer semicontinuous when for every $x_0\in X$ and $\delta>0$ there exists a neighborhood $U$ of $x_0$ such that $F(x)\subset N_\delta(F(x_0))$ for all $x\in U$.

$N_\delta(A)$ stands for a $\delta$-neighborhood of a set $A\subset \mathbb{R}^n$.