Continuity of Lagrange multipliers

Question

Suppose I have an optimization problem of the from $\min f(x)$, s.t. $g(x)=a$, where $f,g$ are real polynomials and $a\in\mathbb{R}$. Then using the Lagrange multiplier rule, I have to find the critical points of the Lagrange function $\mathcal{L}(x,\lambda):=f(x)+\lambda\cdot (g(x)-a)$. Now if I have solved this system and I consider a system that is slightly perturbed, i.e. I replace $f,g$ by $\tilde{f}$ and $\tilde{g}$ which I define as $\tilde{f}_\epsilon:=f+\epsilon\cdot p_1$ and $\tilde{g}_\epsilon:=g+\epsilon\cdot p_2$, where $p_1,p_2$ are fixed polynomials. What can be said about the dependency of the Langrange multipliers of this system on $\epsilon$? Are they also depending continuously an this perturbation?