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December 3rd, 2016, 12:32 PM  #1 
Newbie Joined: Dec 2016 From: Birmingham Posts: 2 Thanks: 0  h(y) is a continuous function of y
I have attached JPG file with the question. Chapter 14 is Integrals on Rectangles and 14.2 is Fubini’s Theorem. But we can not use Fubini’s Theorem because we prove it two problems later in the book. I am completely lost and need help with writing a proof for the problem.

December 5th, 2016, 06:39 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 114 Thanks: 44 Math Focus: Dynamical systems, analytic function theory, numerics 
$f$ is continuous on $R$ so for $\eta = \frac{\epsilon}{ba}$, there exists $\delta$ such that $f(x,y)  f(x,z) < \eta$ if $yz < \eta$. Can you get a bound on $h(y  \delta)  h(y + \delta)$? As a hint, this is given explicitly by \[ \left \int_a^b f(x,y\delta) \ dx  \int_a^b f(x,y+\delta) \ dx \right \] 

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continuous, function, integral, metric spaces 
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