Analytical dependences between first order derivatives at the critical point of a pure substance

The finiteness and positive value of the slope of the saturation curve at the critical point according to its location in the region of a thermodynamic surface restricted by the curves of inversion and saturation are substantiated. It is shown that any two derivatives of the first order formed with respect to volume, internal energy, entropy, enthalpy, and the Helmholtz energy, one of which is taken at a fixed value of pressure and the other at a fixed value of temperature, are equal in the critical state. New sets of the critical conditions for a pure substance are obtained.