What papers or resources explain the conditions needed for filter stability in real-time tuning of parameters?
Instead of trying to fit everything in a comment, I will answer the question here.
Assessing stability varies to which method you use to describe a filter (DE, Laplace domain, State - Space, FIR, etc). Stability itself is a very elaborate and interesting topic in controller and filter design, but I must keep it short enough. In essence, if you apply an impulse response to your filter, the energy of the output must either decrease or stay constant to make the filter (marginally) stable. But this does not tell you how this is linked to the parameters. Depending on your filters description, the following things should be checked for stability:

Laplace domain: roots of the denominator $\leq0$
Discrete Z-domain: absolute value of the roots of the denominator $\leq1$
Continuous State Space: Eigenvalues of A $\leq0$
Discrete State Space: absolute value of the Eigenvalues of A $\leq1$

So, suppose you have a simple LPF in the Laplace domain: $$F(s) = \frac{\omega_0}{s+\omega_0}$$
Then it can be seen that it is stable for any value of $\omega_0\geq0$
Next, lets take a standard FIR system in Z-Domain:
$$H(z) = \frac{b_0 + b_1z^{-1} + ... + b_Nz^{-N}}{1} = \frac{b_0z^{N} + b_1z^{N-1} + ... + b_N}{z^{N}}$$
Since the parameters do not affect the denominator, a FIR filter is always stable!

Is stable to parameter tunings the same as &quot;being continuos function&quot;

If your filter is linear, any continuous input will yield a continuous output, regardless of the parameters used. Any discontinuous input might yield an discontinuous output, but it remains stable. If your filter is non-linear, the methods to assess stability are much harder (Lyapunov equations is one I know), especially to link them directly to parameter tuning (you could trigger a bifurcation point where the filter is stable for some values, but unstable for others).
I hope this clarifies a bit and I am not currently explaining the obvious.