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Increasing functions and their derivatives?
We have a theorem due to Lebesgue that states the following:
If f: [a, b] --> R is an increasing function, then
i) f' exists almost everywhere
ii) f' >= 0 where it exists
iii) there exists a measurable function g such that g = f' almost everywhere
iv) Int_{a,b} f' <= f(b) - f(a)
What I wonder is:
What (if any) additional condition(s) is(are) required to imply that there exists c in (a,b) such that f' is continuous at c?
Further, for any epsilon > 0, is there a function h, continuous on [a,b], such that
Int_{a,b} |h-f'| < epsilon?
Sorry that this question is a bit vague, but I don't know a better way to state it.
I do know that if there is a non-degenerate interval (a',b') in [a,b], then it is true that there must exist such a c (in fact, it will be in (a',b'), and thus in [a,b]).
However, what about functions which are only differentiable almost everywhere?
Further, what about functions that are only differentiable on a set A in [a,b], with A of positive measure in such a set?
1 個解答
- jeredwmLv 61 十年前最愛解答
The answer to your first question is, there really aren't any "reasonable" conditions that will guarantee a continuous f '(x), because there aren't any "reasonable" conditions that f '(x) exists on a whole interval. For instance, you can have a monotone function that is discontinuous on every rational number... As you point out [I think], I think the weakest condition (which is fairly strong) is that f is continuous on a whole interval.
As for your second question, the answer is yes.
Here's the idea... You know there are at most countably many discontinuities in f. Thus, you can take a function which is continuous and within ε/2ⁿ of f near each discontinuity. (I know that's vague, but this is really a bad venue for discussing such constructions! Hopefully you can figure it out.)
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