
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 3rd, 2016, 11:32 AM  #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, 05:39 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 114 Thanks: 45 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 \[ \left \int_a^b f(x,y\delta) \ dx  \int_a^b f(x,y+\delta) \ dx \right \] 

Tags 
continuous, function, integral, metric spaces 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Is the function continuous?  noobinmath  Calculus  6  September 4th, 2015 10:20 PM 
continuous function  Mahboob  Math  1  March 14th, 2015 06:25 AM 
continuous function  sebaflores  Real Analysis  1  October 8th, 2013 11:23 AM 
Is the function continuous at 0  OriaG  Calculus  3  February 9th, 2013 04:35 PM 
continuous function  frankpupu  Calculus  5  February 13th, 2012 03:58 AM 