Find all values of k for which f is uniformly continuous

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

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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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:

Attachment 26521