for m, n being non zero Nat
for f being PartFunc of (REAL m),(REAL n)
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )