Is a the argmin of a strictly convex objective function $g( \boldsymbol{x})$ subject to linear continuously varying constraints continuous?

I have the following problem:

$$\begin{array}{ll} & \boldsymbol{x}^*(t) = \arg \min_{ \boldsymbol{x}}\text{ }g( \boldsymbol{x}) \\ \text{subject to} & \boldsymbol{A}(t)\boldsymbol{x }= \boldsymbol{B}(t)\\ & 0\leq x_i \leq x_{\max} , i=\{1,2,\cdots,N\}\end{array}$$

With $$g(\boldsymbol{x})$$ being a strictly convex function. $$\boldsymbol{x}\in R^{N\times 1}$$. $$\boldsymbol{A}(t) \in \mathcal{R}^{M \times N}$$ has full row rank with $$N>M$$. The elements of $$\boldsymbol{A}(t)$$ and $$\boldsymbol{B}(t)$$ are continuous (possibly smooth if needed) with respect to $$t$$, and a solution $$\boldsymbol{x}^*$$ is always feasible for any time t.

The question now is:

Does strict (possibly strong if needed) convexity of $$g(\boldsymbol{x})$$ this imply that $$\boldsymbol{x}^*(t)$$ is continuous?

PS: I did ask a similar question previously, where I got a good start. However I am not sure if it is enough yet. ( Is the optimal solution of a convex problem continuous with respect to parameters??)

I have now specified the problem a little more, and my goal is to be able to prove that it is so. Any references along with answers is greatly appreciated.

Edit:

Though I am not able to formalize it, my thoughts so far is that I somehow can state and use:

1) The solutions of x has a unique minimum

2) The inequalities should restrict the solutions to a convex hull.

3) The equality gives a solution space, that (or so I believe) when changing smoothly also means that the solutions x* change continuously. Possibly involving the implicit function theorem on the KKT conditions.