let u0 be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in u0,1 holds
ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_partial_differentiable_in u0,1 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume A0: f is_partial_differentiable_in u0,1 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
consider x0, y0, z0 being Real such that
A1: ( u0 = <*x0,y0,z0*> & SVF1 1,f,u0 is_differentiable_in x0 ) by A0, BXXLXSDef6;
thus ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by A1, FDIFF_1:35; :: thesis: verum