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Math Help - Uniform or pointwise continuity?

  1. #1
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    Uniform or pointwise continuity?

    Does  \frac {sinx}{k} converges pointwise or uniformly as k approaches infinity?

    Isn't the limit 0?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Does  \frac {sinx}{k} converges pointwise or uniformly as k approaches infinity?

    Isn't the limit 0?
    converges uniformly on what set? \mathbb{R}? yes it does (to 0). you can use either of the two main definitions to show this. there is an \epsilon definition and a limit definition involving a supremum

    there is also a theorem that has to do with the derivative being bounded implies uniform convergence (or maybe it's uniform convergence implies the derivative is bounded. i can never remember which. i should look it up)
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  3. #3
    Jes
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    Not sure if this argument is good or not. I think it converges to the zero function. It's been a while since I studied calculus.

    If we define the domain as the real line, then we know \sin x is bounded by 1 and \frac {1}{k} \rightarrow 0 for all k \geq N. Let x be in the domain. Given \epsilon > 0, there is some N \in \mathbb{N} such that \left | \frac{1}{k} \right | < \epsilon for all k \geq N. Then  |f_k(x)-0| = \left |\frac {\sin x}{k}\right | = \left|\frac {1}{k} \right ||\sin x | < \epsilon \cdot 1 = \epsilon.
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