1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Where is the following function continuous

  1. Jan 24, 2007 #1
    1. The problem statement, all variables and given/known data
    [tex]f: [0,+\infty) \to \mathbb{R}: y \mapsto \int_0^{+\infty} y \arctan x \exp(-xy)\,dx.
    Show that this function is continuous in [tex]y[/tex] if [tex]y \neq 0[/tex]
    and discontinuous if [tex]y = 0[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I just can't get started, any hint?
  2. jcsd
  3. Jan 24, 2007 #2


    User Avatar
    Homework Helper

    Start by trying to simplify an expression for f(y+d)-f(y). Ultimately you want to show that for any epsilon>0, you can pick a delta so that |f(y+d)-f(y)|<epsilon for all d<delta.
  4. Jan 24, 2007 #3
    Here is my try:
    Choose a sequence [tex]y_n \in [0,+\infty )[/tex] such that [tex]y_n \to y (\neq 0)[/tex].
    Define the function [tex]g_n(x)=y_n \arctan x e^{-xy_n}[/tex], then its limit is [tex]g(x)=y\arctan x e^{-xy}[/tex].
    Note that [tex]|g_n(x)| \leq |y_n\arctan x|[/tex], it follows [tex]g_n[/tex] is integrable. Hence by dominated convergence thm we have
    [tex]\lim f(y_n)=\lim \int g_n \to \int g = f(y)[/tex].

    Am I right? Still no idea for the case [tex]y=0[/tex]
    Last edited: Jan 24, 2007
  5. Jan 25, 2007 #4


    User Avatar
    Homework Helper

    To use the dominated convergence theorem you need to find a function that bounds g_n for all n. In other words, this function can't have an n in it. Other than that you seem to be on the right track. The same idea should work for y=0.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Where is the following function continuous
  1. Continuous Functions (Replies: 2)

  2. Continuity function (Replies: 8)