let z0 be Element of REAL 2; :: thesis: for f being PartFunc of REAL 2, REAL st f is_partial_differentiable`1_in z0 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`1_in z0 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume A0: f is_partial_differentiable`1_in z0 ; :: 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 f,z0 is_differentiable_in x0 ) by A0, Def6;
thus ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by A1, FDIFF_1:35; :: thesis: verum