I have been pondering over something for quite a sometime, and as a part of understanding it, I had to formulate a problem like the one given below. It is quite lengthy and in case it turns out to be totally absurd or trivially wrong, I sincerely apologize for that.
Let $s_n : (0,1) \to \mathbb{R}\quad \forall \; n \in \mathbb{N}$ be a set of smooth functions. The sequence of real numbers $\{s_n(x)\}$ is always positive and increasing for all $x \in (0,1)$. Let $D$ be a countable dense subset of $(0,1)$ and $h : D \to \mathbb{N}$ be an enumeration. The sequence $\{s_n(x)\}$ diverges and $\{s_n(x)\} \in O(\log n) \forall x \in (0,1)\setminus D$The sequence $\{s_n(x)\} \in O(n^{\frac{1}{h(x)}}) \wedge \{s_n(x)\} \in \Omega(n^{\frac{1}{h(x)+1}}) \quad \forall \; x \in D$Additionally $s_n(x)$ is smooth in $(0,1)$ and $s_n(x)$ has finite number of maxima and minima, for all $n \in \mathbb{N}$.
My question is : Is the set of all such sequences nonempty ?
This has been posted on MO as well here