let f be PartFunc of (REAL 3),REAL ; :: thesis: for u0 being Element of REAL 3 st f is_partial_differentiable_in u0,3 holds
SVF1 3,f,u0 is_continuous_in (proj 3,3) . u0
let u0 be Element of REAL 3; :: thesis: ( f is_partial_differentiable_in u0,3 implies SVF1 3,f,u0 is_continuous_in (proj 3,3) . u0 )
assume A0: f is_partial_differentiable_in u0,3 ; :: thesis: SVF1 3,f,u0 is_continuous_in (proj 3,3) . u0
consider x0, y0, z0 being Real such that
A1: ( u0 = <*x0,y0,z0*> & SVF1 3,f,u0 is_differentiable_in z0 ) by A0, BXXLXSDef7ForZ;
SVF1 3,f,u0 is_continuous_in z0 by A1, FDIFF_1:32;
hence SVF1 3,f,u0 is_continuous_in (proj 3,3) . u0 by A1, Th3; :: thesis: verum