let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_partial_differentiable_in z0,1 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 z0,1 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume f is_partial_differentiable_in z0,1 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
then ex x0, y0 being Real st
( z0 = <*x0,y0*> & SVF1 1,f,z0 is_differentiable_in x0 ) by Th5;
hence ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by FDIFF_1:35; :: thesis: verum