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