I am trying to show that a function is upper semicontinuous if and only if the preimage of any open ray $(-\infty, a)$ is open.
The definition given for upper semicontinuity is that $\lim\limits_{k \to \infty} x_k = x \implies \limsup\limits_{k\to \infty} f(x_k) \leq f(x)$.
I find this definition hard to work with, as I have never been comfortable with $\limsup$ and $\liminf$.
Can anyone give me a hint as to how to approach this? Thank you!