Abstract
In elementary calculus, we learn that if a function has a non-negative derivative on an interval, then the function is increasing on that interval. The first result in this chapter weakens the hypothesis on the function and obtains the same result. We then generalize this result with hypotheses involving derivates and approximate derivatives. In Section 5.4, we introduce the Denjoy property: If a function is differentiable everywhere on a closed interval, then the set of points where the derivative is bounded by any two extended-real numbers is either empty or of positive measure. We present Clarkson’s proof of this result followed by a similar result for approximate derivatives. We conclude the chapter with a property related to the Denjoy property; for this last result, we require the concept of metrically dense sets.
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