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My question is that I basically don't understand the whole fifth paragraph of the top-voted answer here: Why is the area under a curve the integral?

How could it be possible for the minimum of the function on one of the split subintervals be greater than or equal to the minimum of the original subinterval before splitting it? Shouldn't it be less than or equal to that minimum? I can't wrap my head around it, even geometrically.

Shocked
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  • When you split the interval, the minimum from before the split might be in one of the subintervals, but not the other one. In any case, the minimum for any of the subintervals can't be less than the minimum on the interval before the split. So if it can't be less, then it must be greater than or equal to. – quasi Jul 20 '17 at 21:32
  • He said that it's possible for the minimum on one of the split subintervals to be larger than the minimum over the whole, original subinterval. How could that be possible? – Shocked Jul 20 '17 at 21:33
  • Take $f(x) = x$ on the interval $[0,2]$. The minimum is $0$, right? Now split the interval as $[0,1],; [1,2]$. What are the minimum values on each subinterval? – quasi Jul 20 '17 at 21:36
  • Thanks. I see now. – Shocked Jul 20 '17 at 21:36

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