I have a a convex program parametrized by the matrix of values $ {\bf v} = \left[v_{i j} \right]_{i \in [m], j \in [n]}$
I define the function $f: \mathbb{R}^m \to \mathbb{R}^n$ which takes as input a vector $\vec{B} = (B_1, \dots, B_i, \dots, B_m)^T$.
This function takes its input $\vec{B}$ and passes it to the convex program below that is parametrized by ${\bf v}$ which solves for the minimizing values of $\forall j, i, \ \ p_j, \ \beta_i$ and returns the vector $\vec{p} = \left( p_1, \dots p_j, \dots, p_n\right)^T$ that minimizes the objective function.
$$ \begin{array}{c} {\min \sum_{j} p_{j}-\sum_{i} B_{i} \log \left(\beta_{i}\right) \text { s.t. }} \\ {\forall i, j, p_{j} \geq v_{i j} \beta_{i}}\\ \text{for variables }\forall i, j \ \ p_j, \beta_i \end{array} $$
My question is this: is there any way for me to prove that the function $f$ is continuous?