Consider the convex optimization problem
$ \min_{x \in X, \ y \in Y } x $
$ \text{sub. to } \ x A + B y + C = 0 $
where $X = [0,1] \subset \mathbb{R}$, $Y \subset \mathbb{R}^M $ are compact and convex sets and $p = (A,B,C)$ is a set of given parameters.
Let $x^*(p)$ the solution associated to parameters $p$.
I wonder if the solution is continuous with respect to the set of parameters:
$ \forall \epsilon > 0 \ \exists \delta>0 \text{ such that: } \ ||p - \tilde{p} || < \delta \ \Rightarrow \ || x^*(p) - x^*(\tilde{p}) || < \epsilon $
In other words, is the solution of a Linear Programming (LP) problem continuous with respect to the parameters of the problem?