Showing utility function gives preferences that are rational and convex 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!
I'll give a few hints to get you started. First, note that since the preference $\succsim$ is represented by the utility function $U(x_1,x_2)=x_1+\ln x_2$, it follows that
\begin{equation}
(x_1,x_2)\succsim (x_1',x_2')\quad\Leftrightarrow\quad U(x_1,x_2)\ge U(x_1',x_2') \tag{1}
\end{equation}
Keeping this equivalence in mind, consider:

Completeness: $\succsim$ is complete if for all $(x_1,x_2),(x_1',x_2')\in\mathbb R_+^2$,
\begin{equation}
\text{either }(x_1,x_2)\succsim (x_1',x_2'),
\quad\text{or }(x_1',x_2')\succsim(x_1,x_2). \tag{2}
\end{equation}
Using $(1)$, we can rewrite $(2)$ as
\begin{equation}
\text{either }U(x_1,x_2)\ge U(x_1',x_2'),
\quad\text{or }U(x_1',x_2')\ge U(x_1,x_2). \tag{2*}
\end{equation}
Now $(2^*)$ should be easy to prove using the property that $\mathbb R$ is an ordered field.

Transitivity: Use the same trick to translate preference ordering into ordering of real numbers.

Convexity: Start from the definition that $\succsim$ is convex if for any $\alpha\in[0,1]$,
\begin{multline}
(x_1,x_2)\succsim(x_1'',x_2'') \text{ and } (x_1',x_2')\succsim (x_1'',x_2'') 
\\\Rightarrow \quad
\alpha(x_1,x_2)+(1-\alpha)(x_1',x_2')\succsim(x_1'',x_2'')
\end{multline}
Again, translate the preference ordering into ordering of real numbers to prove the implication. Since $U$ is quasi-linear, this way will save you some trouble of dealing with Hessians and so on.