Explanation of this hypothesis of SUBSTITUTION RULE requested

My question deals with this following RULE/THEOREM:

The Substitution Rule: If u=g(x) is a differentiable function whose
range is an inteval on which f is continuous, then

integral[ f (g(x)) g' (x) ] = integral[ f (u) du ]


I can't understand why the condition "whose range is an interval on
which f is continuous" is needed. What are some cases that could
break this, and not allow us to use substitution (yet look on the
surface as a good place to use substit.) ?


%%%%%%%%%%
D. Bahar

.

Relevant Pages
Loading