Consider a consumer with preferences relation $\succsim$ over non-negative commodities $x\_1$ and $x\_2$ such that their utility U = $x\_1$ + $\ln(x\_2)$

Are these preferences rational and are they convex/strictly-convex?

I'm a bit confused on how to do this. So first, I know preferences need to be complete and transitive to be rational, but for a utility function, it just needs to be continuous right? Is there another property it needs? With that said, how would I mathematically prove this function is actually continuous? If I graph it, it's continuous, but is there a mathematical proof for this?

For the second part, if preferences are (strictly) convex, then the preferences must be (strictly) quasi-concave right? How would I mathematically prove that this function above is quasi-concave?

Online, it says a function is quasi-concave if $f(\lambda x+(1-\lambda )y)\geq \min {\big \{}f(x),f(y){\big \}}$, but I'm having a tough time understanding this in relation to a utility that has both an $x\_1$ and an $x\_2$ value. When I'm looking at the above function, I only understand it for like $f(a) = a^2$ and there's not a second variable in there.

Thanks!