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rsa58
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Homework Statement
how do you show that sin(1/x) is continuous on (0,1)? (i know it's also continuous on (0, infinite)).
Homework Equations
The Attempt at a Solution
|f(x)-f(xo)| = |sin(1/x)- sin(1/xo)|= |2sin((xo-x)\2)cos((xo+x)/2)|
=< 2|sin((xo-x)/(2xox))|=< |(xo-x)/(xox)|. is this inequality true? a similar one is used in a different example. if it is, why? is it because sin(x)=<x ? when x is positive? now since x<1 choosing [tex]\delta[/tex]=xo[tex]\epsilon[/tex] then if
|xo-x|<[tex]\delta[/tex] then |f(xo)-f(x)|<[tex]\epsilon[/tex]
is this answer correct? what about the endpoints?