when does integration preserve uniqueness of a minimization problem?

Question

Suppose $X\in[0,1]$ and for every $X=x$, let $y(x)=\arg\min_{z} f(z;x)$. For every $X=x$ the minimizer $y(x)$ is unique. Let $G(\cdot)$ be the distribution of $X$. I want to find a minimizer y* for $\int f(z;x)dG(x)$. Under what condition on $f(\cdot;\cdot)$ and $G(\cdot)$ can we guarantee that the solution $y*$ is unique?