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!

Prove Sin(x):R->R is continuous

  1. Sep 26, 2009 #1
    Prove Sin(x):R--->R is continuous

    1. The problem statement, all variables and given/known data

    Prove that Sin(x) from R to R is continous using the epsilon delta definition of continuity and the following lemma:

    denote the absolute value of x by abs(x)
    Lemma: abs(x)>=sin(x)

    2. Relevant equations

    Could somebody please just tell me if my proof is correct?
    The only possible problem that I see is here:
    abs(sinp - sinq)=<abs[abs(p)-abs(q)]
    thank you

    3. The attempt at a solution

    p,q are any points in R
    denote the distance between p and q by d(p,q)

    for each epsilon>0 choose 0<delta<epsilon
    d(p,q)=abs(p-q)<delta

    d(sinq,sinq)=abs(sinp - sinq)=<abs[abs(p)-abs(q)]=<abs(p-q)<delta<epsilon
     
  2. jcsd
  3. Sep 26, 2009 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Prove Sin(x):R--->R is continuous

    The only possible problem that I see is here:
    abs(sinp - sinq)=<abs[abs(p)-abs(q)]

    That is a problem since for p = 2, q = -1 it gives:

    1.750768412 < 1
     
    Last edited: Sep 26, 2009
  4. Sep 27, 2009 #3

    lurflurf

    User Avatar
    Homework Helper

    Re: Prove Sin(x):R--->R is continuous

    you want to show
    |sin(x+h)-sin(x)|<epsilon
    |sin(x+h)-sin(x)|=|cos(x+h/2)||sin(h/2)/(h/2)||h/2|
    take it from there
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook