An example of a function $f ∈ L_1(\mathbb{R})$ such that $\limsup_{x→\infty} f(x) = \infty$ I have come with an example of such a function which I'm a little unsure of. My course doesn't cover the Lebesgue integration very well, so I am trying to clarify my doubts here.
My function is $f(x)= n$ whenever $x=n\in\mathbb{N}$ and $0$ otherwise. Clearly, $\limsup_{x→\infty} f(x) = \infty$ but how do I show this function is in $L_1(\mathbb{R})$ with rigour? Any ideas?

In $f(x)$, u defined, $f(x)≠0$ in a set of measure $0$. So wont affect the integration and hence $L_1(R)$.
