theorem Th45: :: PDIFF_9:45

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