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An unusual monotonicity theorem with applications. (English) Zbl 0651.26006

The authors prove the following monotonicity theorem (\(\mu\) * - the outer Lebesgue measure): Suppose that f is a real-valued function defined on an interval [a,b] and that \(\alpha\in (0,1)\). Suppose further that there is a set \(S\subset (a,b)\) such that the following conditions hold.
(A) For every \(x\in (a,b)\setminus S\), there is \(y\in (x,b]\) such that \(\mu \quad *\{z\in (x,y):\quad f(z)>f(x)\}\geq \alpha (y-x),\) or there is \(y\in [a,x)\) such that \(\mu \quad *\{z\in (y,x):\quad f(z)>f(x)\}<\alpha (x-y).\)
(B) For every \(x\in [a,b)\) and every \(c<f(x)\), there is \(y\in (x,b]\) such that \(\mu \quad *\{z\in (x,y):\quad f(z)>c\}\geq \alpha (y-x).\)
(C) For every \(x\in (a,b]\) and every \(d>f(x)\), there is \(y\in [a,x)\) such that \(\mu \quad *\{z\in (y,x):\quad f(z)>d\}<\alpha (x-y).\)
(D) f(S) contains no interval.
Then f(b)\(\geq f(a).\)
As consequences can be obtained some usual monotonicity theorems which deal with preponderant and symmetrical derivatives. E.g., it is shown, that if f is an approximately continuous function on (a,b) such that \[ \underline D_{sym}f(x)\quad =\quad \liminf_{h\to 0+}(f(x+h)-f(x- h))/2h\quad \geq \quad 0 \] except a countable set, then f is nondecreasing on (a,b). This answers a question posed by L. Larson [Real Anal. Exch. 9(1983-84), 295 (1984), Query 166].
Reviewer: P.Kostyrko

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A48 Monotonic functions, generalizations
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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