Let and be continuous functions. Show that defined by , is continuous as well.

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- Apr 4th 2009, 08:40 AMAndreametContinuity with product of the range
Let and be continuous functions. Show that defined by , is continuous as well.

- Apr 4th 2009, 09:45 PMaliceinwonderland
- Apr 6th 2009, 04:44 AMxalk
**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