I have a very simple class of optimization problems. The objective function is separable with each part being strictly (or even strongly) convex. Furthermore, there's a solitary linear constraint: the sum of the variables is $b$. The variables must all be non-negative.
Can we say that the optimal solution, denoted $x(b)$, is a continuous function of the scalar $b$?
minimize $\sum f_i(x_i)$
subject to: $\sum x_i = b$
with: $x_i \ge 0.$
If necessary, assume that there exists $m>0$ such that $f_i''(t) > m$ for all $i$ and for all $t \ge 0$.