Let and be continuous functions. Show that defined by , is continuous as well.
Let ε>o and aεX.
Since and
,then there exist:
and such that:
if and xεX ,then <ε/2 for all,x............................................. .......................................1
and such that:
if and xεX, then <ε/2 for all ,x................................................ ...........2.
Choose = min{ }
Let |x-a|<δ and xεX.
then and and by (1) and (2) we have:
BUT.
Norm (h(x)-h(a)) = ||h(x)-h(a)|| =
Thus ,for all ,a in X AND hence the function ,h is continuous over X