As an example, mechanical work and heat are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.
Examples of path functions include work, heat and arc length.
In other words, for a holonomic process function, λ may be defined such that dY = λδX is an exact differential.
For example, thermodynamic work is a holonomic process function since the integrating factor λ = 1/p (where p is pressure) will yield exact differential of the volume state function dV = δW/p.
The second law of thermodynamics as stated by Carathéodory essentially amounts to the statement that heat is a holonomic process function since the integrating factor λ = 1/T (where T is temperature) will yield the exact differential of an entropy state function dS = δQ/T.