To the mean-value theorem (Q1945782)

The author proves several extensions of the well-known Lagrange mean value theorem for cases of continuous functions on the real line and in the complex plane. The paper starts with integrating (Denjoy) the equations in the Lagrange mean value theorem and recognizing that the slope of the chord through \((a,f(a))\) and \((b,f(b))\) is equal to the average value of the derivative. The author then proceeds to consider the sets \(\left\{ \frac{f(y)-f(x)}{y-x}\right\} \) and \(\left\{ f^{\prime}(x)\right\} \) and restating the theorem with their help. Next, the behavior of the function \({\mathbf \Phi}(x,y)=(\Phi_{1}(x,y),\Phi_{2}(x,y))\) with \(\Phi_{1}(x,y)=x\) and \(\Phi_{2}(x,y)=\frac{f(y+\delta)-f(y)}{\delta}-f^{\prime}(x)\) is employed to obtain uniqueness results for the Lagrange point. This results is related to the convexity of an arc. The paper then considers Lagrange second-kind points, those are points where one mean value is attained at the endpoint of the interval under consideration; the author then gives a condition of the convexity of a certain type of arc in terms of the lack of such second-kind points and the existence of inflection points is shown to be equivalent to existence of second-kind points. A further section is dedicated to the study of derivative numbers, (the limit of a difference quotient along a sequence), which allows the author to extend results to non-differentiable functions via the study of contingencies and intermediate semi-tangents. The paper concludes with a look at functions in the complex plane and a study of what results can be carried over.