Re: riemann integrability


abalone wrote:
> let f : R -> R. f is bounded and as h->0, lim f(x+h) exists for all
x
> in R. Show that f is Riemann integrable on any bounded closed
interval.

Here is a hint.Let g(x)=lim f(x+h) (h->0) .1)show g is continuous at
every x. 2) On any bounded closed interval ,and any positive integer
n,the set of points x with |f(x)-g(x)|> 1/n is finite. From this it
should be easy to show that the limit of the Riemann sums,or the upper
and lower integrals of f-g are all 0.Since f=(f-g)+g ,g continuous,it
follows that f is Riemann integrable with the same integral as g.
Where are these interesting problems coming from? Regards again,Stuart
M Newberger

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