measure theory - A function that is not Lebesgue integrable on [0,1]

I'm currently studying for my measure theory final, and I am struggling with a question:

Give an example of a Borel-measurable function $X : (0,1) \to \mathbb{R}$ that satisfies:

For every $a \in (0,1)$, the function $X$ is Lebesgue-integrable on $[a,1]$,

The limit $\lim_{a\to 0} \int_{a}^{1}X d\lambda$ exists and is finite, with $\lambda$ the standard Lebesgue measure,

The function X is not Lebesgue-integrable on $(0,1]$.

I hope someone could help me out,

thanks.