let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_partial_differentiable_in z0,2 holds
ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( f is_partial_differentiable_in z0,2 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume A0: f is_partial_differentiable_in z0,2 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
consider x0, y0 being Real such that
A1: ( z0 = <*x0,y0*> & SVF1 2,f,z0 is_differentiable_in y0 ) by A0, Def7;
thus ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by A1, FDIFF_1:35; :: thesis: verum