In the famous TKNN paper and subsequents the authors wisely argue that the transversal conductance in the Integer Quantum Hall Effect has a topological interpretation as the integral of the curvature of the Berry connection on the Brillouin zone which happens to coincide with the first Chern number for a $U(1)$ principal bundle on a manifold. In particoular Kohmoto justifies the quantization of the TKNN invariant with the following argument:

let's $\psi_{\mathbf{k}} ( \mathbf{x} )$ be the wave function in the Brillouin zone associated with $\mathbf{k}$ quasi-momentum. He choses a section of the bundle such that $\psi_{\mathbf{k}} ( \mathbf{x}_0 )$ at fixed $\mathbf{x}_0$ is real. This of course doesn't work at the points where the wavefunction nullify. So he encircles them and defines the connection $A$ smoothly inside and outside the circles, with different prescriptions. He then applies Stoke's theorem to this situation getting a sum of line integral of the difference between the connection inside and outside the circles encompassing the singularities:

$$

\sigma_{xy} = \sum_{i} \int_{\partial S_i} ( \mathbf{A}_{out} - \mathbf{A}_{in} ) \cdot \mathbf{d l}

$$

The difference of the connections is due to the discontinous choice of the phase:

$$

\psi_{\mathbf{k}}^{out} = e^{i f(\mathbf{k})} \psi_{\mathbf{k}}^{in}

$$

$e^{i f(\mathbf{k})}$ needs to be single valued but $f(\mathbf{k})$ needn't! $e^{if}$ is basically the transition function that defines the principal bundle. If the function $f$ is differentiable then we can write a gauge transformation for the potential $A$ which looks like:

$$

\mathbf{A}_{out} = \mathbf{A}_{in} + \nabla f

$$

then the total conductivity becomes:

$$

\sigma_{xy} = \sum_{i} \int_{\partial S_i} ( \nabla f) \cdot \mathbf{d l}

$$

now the requirement for the phase difference to be well defined fixes each integral to be $2 \pi n$.

I've alway forgetten constants in my derivations, so don't mind the extra $2 \pi$ that happears. Here we come to my questions:

Is the smoothness hypothesis on $f$ well motivated? I think so because one can smoothly choose the phases of the wave function in the inner and outer regions. Then the difference will be smooth.

This arguments shows the quantization of the integral but don't show the stability under smooth variation of $\psi_{\mathbf{k}}$ which I expect indeed to be true. Can we see it with basic notions? Basically I want to understand how the twists in the wavefunction are protected against variations of the Hamiltonian.

Please, I don't know advanced maths and I cannot follow real mathematical general motivations one can find on topology books. I was looking for a more down to earth justification.

Thanks in advance!