On bilateral derivates and the derivative

Abstract:

In this paper we prove a new result on the monotonicity of a function in terms of its bilateral derivates, and obtain from it extensions of several existing results on such derivates and the derivative of a function. Let $f:R \to R$, where $R$ denotes the set of real numbers. If its lower derivate $\underline {D} f > 0$ at a nonmeager set of points, we prove $f$ to be “adequately” increasing in some interval, viz. even the function $f(x) - \alpha x$ is increasing for some $\alpha > 0$. When $f$ is nowhere adequately monotone, it follows that there exists a residual set of points where $f$ has a zero “median” derivate, i.e. either $D\_f \leqslant 0 \leqslant {D^ - }f$ or ${D_ + }f \leqslant 0 \leqslant {D^ + }f$. These results remain valid for functions defined on an arbitrary set $X \subset R$ under a mild continuity hypothesis, e.g. the absence of ordinary discontinuity at the unilateral limit points of $X$. The last result leads to a new version of A. P. Morse’s theorem for median derivates, and this in turn yields an improved version of the Goldowski-Tonelli theorem. We also obtain some necessary and sufficient conditions for a function to be nondecreasing, and extensions of the mean-value theorem and the Denjoy and other properties of a derivative. If $f:X \to R$, where $X \subset R$, and both the derivates of $f$ are finite at a set of points that is not meager in $X$, then $f$ is further proved to satisfy the Lipschitz condition on some portion of $X$. When $f$ has a finite derivate almost everywhere and $X$ has a finite measure, it is shown that $f$ can be made Lipschitz by altering its values on a set with arbitrarily small measure. Some results on singular functions are also strengthened. The results and the methods of this paper further provide extensions of some results of Young, Tolstoff, Kronrod, Zahorski, Brudno, Fort, Hájek, Filipczak, Neugebauer and Lipiński on derivates and the derivability of a function.