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Math Help - Find all values of k for which f is uniformly continuous

  1. #1
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    Find all values of k for which f is uniformly continuous

    Find all k \in R such that f= x^{k} is uniformly continuous, x \in (0, + \infty )

    I can check if a certain function is uniformly continuous, but I don't know how to check for what values it is uniformly continuous.

    Please, help.
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  2. #2
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    Re: Find all values of k for which f is uniformly continuous

    The answer is all k in the closed interval [0,1]. Informally, the reasoning is as follows. For k > 1, and large values of x the derivative is quite large and so a small change in x values produces large change in the function values. Similarly for k < 0 and x values close to 0. For k between 0 and 1, the function is well behaved on [0,1] and its graph gets quite flat for large values of x.

    Here is "2/3" of the actual proof:
    Find all values of k for which f is uniformly continuous-uniformcontinuity.png
    Thanks from wilhelm
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