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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 :)

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?

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....

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.

Re: Question on proving the property of an integral

yep, made sense of it. Thanks for your help :)