A positive function $u$ such that $\liminf_{y\to x}u(y)=0$ for all $y$ but $u\neq 0$ a.e.?

Question

As stated above, does there exist a $u$ on a domain $D\subset\mathbb{R}^n$ such that $\liminf_{y\to x}u(y)=0$ for all $x\in D$ but $u$ is not vanishing almost everywhere?

It seems so, but I hope not. However I know there are some weird counterexamples in real analysis which will likely put my hopes to rest.

I thought of the function $1-\chi_{\mathbb{Q}}$ on $\mathbb{R}$, but this is not an everywhere positive function.

Any help would be greatly appreciated. Thanks

Answer 1

Let $X = \{x_1, x_2, \cdots \}$ be a countable dense subset of $D$, and define $u : D \to \mathbb{R}$ by

$$ u(x) = \begin{cases} 1/n, & \text{if } x = x_n \\ 1, & \text{otherwise} \end{cases}. $$

Then for each $x \in D$ you can extract a sequence $(x_{n_k})$ such that $0 < |x - x_{n_k}| \to 0$, hence

$$0 \leq \liminf_{y\to x} u(y) \leq \lim_{k\to\infty} \frac{1}{n_k} = 0. $$