Of course. Entropy has a discontinuous jump at a first order transition but changes continuously at a second order one.

Entropy doesn't change at a second order phase transition-it changes at a first order phase transition. That change is known as the latent heat, which is zero in the former case and non-zero in the latter.

During the first order phase transition the first order of derivatives from Gibbs free energy are discontinious and for the second order phase transition only the second derivatives are discontinius. Becouse the entropy of such processes is the first derivative from the temperature with the opposite sign, in the first case difference in entropy (which hops between phases) can be described by latent heat of vaporization/condensation at the some surface, which in function of equlibrium value of this surface temperature only. In the secound case in some processes (such as melting/hardening of alloys) we can determine only approximate quantities of phases change in equilibrium state for appropriate critical temperature.

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity

In terms of Gibbs free energy, which is one of the four thermodynamic potentials, one may look at this question in the following way. If G versus T is plotted for constant (P,N) then, for the first order, the curve is not differentiable at the point which corresponds to the value of T at which phase transition is taking place. As entropy is the negative of the slope of tangent of this curve, it undergoes an abrupt change, characterizing that the phase transition is of first order. In a second order transition, the G vs T curve is both continuous as well as differentiable, making the change in entropy a continuous process. If second derivative also exists then you will get a third order phase transition and so on.

In the language of Statistical Mechanics, entropy is related with the number of microstates accessible to a given set of macroscopic parameters. In first order phase transition, number of available microstates suddenly increases at a particular temperature, whereas in a second order transition this change is smooth and gradual.

Thermodynamic potential G is obtained from a derivative of the partition function Z, thus it is important to study analyticity properties of the partition function in the thermodynamic limit in order to determine order of phase transitions.

Calculus is not "take for granted".

ΔQ/T can not be turned into dQ/T. That is, the so-called "entropy " doesn't exist at all.

It is well known that calculus has a definition.

Any theory should follow the same principle of calculus; thermodynamics, of course, is no exception, for there's no other calculus at all, this is common sense.

Based on the definition of calculus, we know:

to the definite integral ∫T f(T)dQ, only when Q=F(T), ∫T f(T)dQ=∫T f(T)dF(T) is meaningful.

As long as Q is not a single-valued function of T, namely, Q=F( T, X, …), then,

∫T f(T)dQ=∫T f(T)dF(T, X, …) is meaningless.

1) Now, on the one hand, we all know that Q is not a single-valued function of T, this alone is enough to determine that the definite integral ∫T f(T)dQ=∫T 1/TdQ is meaningless.

2) On the other hand, In fact, Q=f(P, V, T), then

∫T 1/TdQ = ∫T 1/Tdf(T, V, P)= ∫T dF(T, V, P) is certainly meaningless. ( in ∫T , T is subscript ).

We know that dQ/T is used for the definite integral ∫T 1/TdQ, while ∫T 1/TdQ is meaningless, so, ΔQ/T can not be turned into dQ/T at all.

that is, the so-called "entropy " doesn't exist at all.

About glass transition, which has been thought as somehow resembling a second order phase transition in terms of the Ehrenfest classification, even if this analogy is based at idealized circumstances that can not be realized in practice; cf.:

Article Aspects of the non-crystalline state