Example of a function $f > 0$ with $f'\leq 0$ but $\nexists \lim_{t\to \infty} f'(t)$

Question

This question is related to this Wikipedia article about Lyapunov theory and Barbalat's lemma. Consider a function $f: \mathbb{R} \to \mathbb{R}$ at least two times differentiable with the following properties:

1. $ f(t) > 0$ for all $t$

2. $f'(t) \leq 0$ for all $t$.

Then the function converges $\exists \lim_{t \to \infty} f(t)$. But is is known that this is not enough to assert the that $f'(t)$ converges to zero. (uniform continuity of $f'$ is required).

Question:

Can some one provide an example of such a function, with properties 1) and 2) but for which $f'(t)$ does not converge to zero?

My try:

My impression is that $f'$ besides being negative has to have some "bumps" which are getting narrower as $t \to \infty$ while keeping the magnitude above a certain threshold ...

Answer 1

Let's consider a function $h :\mathbb{R} \to \mathbb{R}$ with the following properties: $h\in C^1$, $\int_0^1 h(t)dt=1$ and $h=0$ if $t \not \in [0,1]$ (Such functions are easy to construct).

Then, you can define $f$ to be a function such that $f'(t) = -h(\frac{1}{2^n}\cdot (t-n))$ if $t \in [n, n+1]$, and $f(0)=2$. It is then easy to see that $f'(t) < 0$ and $f(n)$ = $\frac{1}{2^n}>0$ so that $f> 0$ and $f' \leq 0$ and $f'$ doesn't have a limit at $+ \infty$

Answer 2

Let $$ g(x)=\max\{0,\sup\{\,1-2^n|x-n|:n\in\Bbb N\,\}\}$$ and $$f(x)=2-\int_0^x g(x)\,\mathrm dx.$$