theorem Th23: :: MESFUN16:23

for I, J being closed_interval Subset of REAL
for f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
for g being PartFunc of [:REAL,REAL:],REAL st f is_continuous_on [:I,J:] & f = g holds
for e being Real st 0 < e holds
ex r being Real st
( 0 < r & ( for x1, x2, y1, y2 being Real st [x1,y1] in [:I,J:] & [x2,y2] in [:I,J:] & |.(x2 - x1).| < r & |.(y2 - y1).| < r holds
|.((|.g.| . [x2,y2]) - (|.g.| . [x1,y1])).| < e ) )