:: deftheorem defines is_uniformly_continuous_on NFCONT_2:def 2 :
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_uniformly_continuous_on X iff ( X c= dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((f /. x1) - (f /. x2)).| < r ) ) ) ) );