Continuous increasing function with different Dini derivatives at 0

Is there an increasing continuous function $f\colon[0,1]\to\mathbb{R}$ such that the right upper derivative $D^+f(0)$ does not equal the right lower derivative $D_+f(0)$?

Recall: $$D^+f(0)=\limsup_{h\downarrow0}\frac{f(0+h)-f(0)}{h},\quad D_f(0)=\liminf_{h\downarrow0}\frac{f(0+h)-f(0)}{h}. $$

Solution 1:

Consider the piecewise linear function pictured below

It starts at $(3,6)$ goes to $(2,1)$ then on to $(\frac14,\frac12)$. Repeat this at scales of $1:12$ and we get a continuous, monotonic increasing function that has an upper derivative of $2$ and a lower derivative of $\frac12$ at $0$.

Smooth example:

Using the idea of the piecewise linear function above, I came up with $$ f(x)=\frac{x}{4}(5+3\sin(\log(x)) $$ $\dfrac{f(x)}{x}$ also bounces between $\frac12$ and $2$ (like the piecewise linear function above) and $$ \begin{align} f'(x)&=\frac54+\frac34\sin(\log(x))+\frac34\cos(\log(x))\\ &=\frac54+\frac{3\sqrt{2}}{4}\sin\left(\frac{\pi}{4}+\log(x)\right)\\ &\ge\frac{5-3\sqrt{2}}{4}\\ &>0 \end{align} $$

Solution 2:

The following is an edited excerpt from a sci.math post (made on 3 November 2000) that may be of interest. The original post (3 URLs are provided below to better guard against link rot) contains quite a bit more information about the possible Dini derivate behavior of monotone functions.

http://groups.google.com/group/sci.math/msg/1bd39d992c91e950

http://mathforum.org/kb/message.jspa?messageID=274836

https://groups.google.com/forum/?fromgroups#!msg/sci.math/8vp1XK0-KNg/UOmRLJmd0xsJ

Zamfirescu [2] [3] proved that most nondecreasing continuous functions on $[0,1]$ are not differentiable (finitely or infinitely) at most points, when most is used the Baire category sense in both places. In fact, he proved that most nondecreasing continuous functions on $[0,1]$ have the behavior you're asking about at most points. More precisely, Zamfirescu proved the following.

Let $C^{inc}[0,1]$ be the collection of nondecreasing continuous functions with the sup norm. Then, for each $f \in C^{inc}[0,1]$ except for a first category set of functions $f,$ we have:

(1) $D_{-}f(x) = D_{+}f(x) = 0$ and $D^{-}f(x) = D^{+}f(x) = \infty$ for each $x \in [0,1]$ except for a first category set of points in $[0,1].$ [3]

(2) $D_{-}f(x) = 0$ or $D^{-}f(x) = \infty$ for each $x \in (0,1].$ [2]

(3) $D_{+}f(x) = 0$ or $D^{+}f(x) = \infty$ for each $x \in [0,1).$ [2]

(4) $f'(x) = 0$ for each $x \in [0,1]$ except for a Lebesgue measure zero set of points in $[0,1].$ [2]

Note that (2) and (3) taken together imply that at each point there does not exist a positive finite unilateral derivative. F. S. Cater gives a nice construction of a function that satisfies (2) and (3) taken together in [1].

[1] Frank Sydney Cater, On the Dini derivates of a particular function, Real Analysis Exchange 25 (1999-2000), 943-946.

http://tinyurl.com/88kcze2

[2] Tudor Zamfirescu, Most monotone functions are singular, American Mathematical Monthly 88 #1 (January 1981), 47-49.

http://tzamfirescu.tricube.de/TZamfirescu-078.pdf

[3] Tudor Zamfirescu, Typical monotone continuous functions, Archiv der Mathematik 42 #2 (1984), 151-156.

http://tzamfirescu.tricube.de/TZamfirescu-087.pdf