Let $f:[1,+\infty)\rightarrow\mathbb{R}$ be a concave function such that $ \displaystyle{\lim_{n\to\infty}}(f(n)-f(n+1)=0$ and the derivative of the function $f$ does not exist. I am looking for an example of a two-rule function that in addition to the above conditions.
My attempt: Define the function $f:[1,+\infty)\rightarrow\mathbb{R}$ by \begin{align*} f(x)= \begin{cases} \log x &; x\geq 2 \text{}\\ \frac{2}{3}x + \log 2 - \frac{4}{3} &; 1\leq x < 2 \text{} \end{cases} \end{align*} This function is concave and $ \displaystyle{\lim_{n\to\infty}}(f(n)-f(n+1))=0$ from a number on. There are two problems in this example: First, the function does not have a left-hand limit at $x=2$ . Secondly, $ \displaystyle{\lim_{n\to\infty}}(f(n)-f(n+1))=0$ is from a number on, whereas I want $ \displaystyle{\lim_{n\to\infty}}(f(n)-f(n+1)=0$ in $[1,+\infty)$. Can anyone guide me and give an example with the above conditions?. Thanks