uniform convergence on union interval

Theorem. If  a<b<c  and the sequencef1,f2,f3,  of real functions converges uniformly both on the interval  [a,b]  and on the interval  [b,c],  then the functionMathworldPlanetmath sequence converges uniformly also on the union (http://planetmath.org/Union) interval  [a,c].

Proof. We have the limit functions fab:=limnfn  on  [a,b]  and  fbc:=limnfn. It follows that


Define the new function


Choose an arbitrary positive number ε. The supposed uniform convergencesMathworldPlanetmath on the intervals  [a,b]  and  [b,c]  imply the existence of the numbers n1(ε) and n2(ε) such that




If one takes  n>max{n1(ε),n2(ε)},  then one has simultaneously on both intervals  [a,b]  and  [b,c],  i.e. on the whole greater interval  [a,c],  the condition

Title uniform convergence on union interval
Canonical name UniformConvergenceOnUnionInterval
Date of creation 2013-03-22 17:27:09
Last modified on 2013-03-22 17:27:09
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 40A30
Related topic MinimalAndMaximalNumber