# Question on proving the property of an integral

• May 24th 2012, 08:47 AM
darkPassenger
Question on proving the property of an integral
Hi,

I have a question which I'm having trouble with.

Attachment 23941
I am told that I should first use the epsilon delta definition of continuity to prove that there must be a closed interval [c,d] with c < x0 < d such that for some x in [c, d], |h(x)| > |h(x0)|/2 by choosing epsilon to be |h(x0)|/2. I am having trouble with this.

Any help would be appreciated :)
• May 24th 2012, 09:12 AM
emakarov
Re: Question on proving the property of an integral
Just instantiate epsilon to |h(x0)| / 2 in the definition of continuity of h at x0. What do you get?
• May 24th 2012, 09:22 AM
darkPassenger
Re: Question on proving the property of an integral
Errr....well h is continuous at x0 when if |x - x0| < delta, then ||h(x)| - |h(x0)|| < |h(x0)|/2. Sorry but you might have to spell it out for me....
• May 24th 2012, 09:36 AM
HallsofIvy
Re: Question on proving the property of an integral
Which, in turn, means that -h(x0)/2< h(x)- h(x0)< h(x0)/2. (You don't need absolute value on h(x0) because they are by hypothesis positive) and the absolute values on h(x) and h(x0) on the left are wrong.)

Now add h(x0) to each part. h(x0)- h(x0)/2= h(x0)/2< h(x)< h(x0)/2+ h(x0)= 3h(x0)/2. And, again, by hypothesis h(x0) is positive.
• May 24th 2012, 09:48 AM
darkPassenger
Re: Question on proving the property of an integral
yep, made sense of it. Thanks for your help :)