Do we impose that the scale factor $a(t)$ of the Universe is a continuous function? Or there is a physical meaning?
Usually in physics we define functions to be continuous, such as the velocity of a point particle of mass m, but is this acceptable for all physical scales($10^{-10}$m to $10^{10}$m for example)?
The current form of the cosmological scale factor is continious function and can be precisely derived. See for example https://arxiv.org/abs/1109.2258
You may explore a connection to the Friedman equations in order to understand the significance of the scale factor as a function of time. Basically, it dictates the dynamic of the Universe, or (as some say) vice versa. The scale factor should be applicable to all distances for all scales while opinions may differ here, as the scale factor is not studied at the micro-scales.
The Friedmann equations are differential equations: insofar they are a correct description of what happens expansion and contraction have to be continuous. Were $a(t)$ to discontinuously change value they would become ill-posed and not predict what the universe does next.
In fact, deep down General Relativity is based on the idea that space-time is a metric manifold $(M,g)$ where the metric is $g$ differentiable and the manifold $M$ locally always can be mapped to Minkowski space. Discontinuous jumps of $a(t)$ implies nondifferentiability of $g$, and were it to correspond to more than coordinate shenanigans then it would break the manifold topological structure.
Over what range this model is applicable remains to be seen, but at least large-scale it seems to be doing fine.
