Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE:

$\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 \phi(t,S,i)}{\partial S^2}$ + $\sum_{j=1}^k \lambda_{ij} \phi(t,S,j) = r_i \phi(t,S,i)$

with boundary conditions: $\phi(T, S, i) = max(S - K, 0)$ (K is a constt), and $\phi(t,0,i)=0$.

$\lambda$ is a k by k square matrix (we can go ahead and say that it's a rate matrix) and $t \in [0,T]$.

Now let $x_{\lambda}$ be the classical solution for the PDE. I've checked that the solution does exist (but I don't know it's exact form). It is, of course, a function of the parameters involved in the PDE, including $\sigma$.

I'm trying to see if I can figure out $\sigma$ if $x_{\lambda}$ is known, for which I'll need to use implicit function theorem.

Now the problem is:

I wish to check for (more like prove) the continuity of these maps: $\lambda \mapsto x_{\lambda}$, and

$x \mapsto \frac{\partial x}{\partial \sigma}$.

I've spent hours on this and I can't figure out how to show that these maps are continuous. I thought I had a way to do this for the second map, but now I've hit a dead end, and I think I need help.

Thank you

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