Re: Explanation of this hypothesis of SUBSTITUTION RULE requested


deniz.bahar@xxxxxxxxx wrote:
Arturo Magidin wrote:
In article <kx8Kf.3041$%14.96328@xxxxxxxxxxxxxxxxxxxxx>,
True Raptor <CB4ever@xxxxxxxxxxxxxxx> wrote:
deniz.bahar@xxxxxxxxx wrote:
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.) ?


If g is defined for some values for which f is not continuous, then the
integral does not exist.

May fail to exist.

Integration requires continuity.

No, it does not require continuity. Continuity guarantees the
existence of the integral, but is not required. A step function, for
example, is easily shown to be integrable even if it is not exontinuous.



so 'u = g(x)' doesn't need to be defined for a range where f is purely
continuous?

I was addressing the other poster's assertion that "integration
requires continuity"; that is simply not the case. I addressed your
question elsewhere. The condition given in your book a bit stronger
than strictly needed, but they are much easier to state than "tight"
conditions would be. In practice the situation described in the book is
the one that usually arises, so it suffices. It would be overly
confusing for your book to attempt to provide a very precise set of
conditions, since they usually involve a number of cases and
situations. Then it would be unwieldly.

I guess the book went overboard by mentioning this
condition.

No. It just gave you the simplest and most common situation; without
that hypothesis, you need a number of other things to work out properly
before you can guarantee that the two integrals will agree on a given
interval.

Arturo Magidin, sans .sig

.

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