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prove that if the function e^x * sin (1/x) is uniformly continuous on the interval...

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cica777 | Student, College Freshman | eNotes Newbie

Posted April 12, 2012 at 10:56 PM via web

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prove that if the function e^x * sin (1/x) is uniformly continuous on the interval a) (0,1) b) [1/2, 3/2]? 

I also know that this function is continuous on (o,infinity) but I don't understand why...

Tagged with continuity, fuction, math

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beckden | High School Teacher | (Level 1) Educator

Posted April 13, 2012 at 5:36 AM (Answer #1)

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e^x is uniformly continuous on (-oo,oo).

1/x is uniformly continuous on (-oo,0) U (0,oo).

sin(x) is uniformly continuous on (-oo,oo) this implies sin(1/x) is uniformly continuos on (-oo,0) U (0,oo).

If f(x) and g(x) are uniformly continuous on an interval f(x)*g(x) is uniformly continuous on that same intervarl,
so e^x sin(1/x) is uniformly continuous on (-oo,0) U (0,oo)
 so it is certainly uniformly continuous on subintervals of those intervals.

since (0,1) and (1/2, 3/2) are subintervals of (0,oo) e^x sin(1/x) is uniformly continuous on those intervals also.

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