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This is a problem that I had already stumbled upon Riemann integrals anc choosing partitions. Given a function and asked to prove that is Henstock-Kurzweil integral by definition, how do you find the appropriate gauge?

For example, given $$f(x)= \frac{1}{\sqrt{x}}$$ In a lot of textbooks the gauge is just arbitrarily given. While I assume some cleverness and guessing helps, I guess there is at least some logic that lead to that particular gauge. So say, in the example given above, how would you choose/construct the gauge to prove that is integrable?

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