Are monotonic and continuous preferences necessarily rational?
Consider a preference relation in $\mathbb{R}^2$ such that $x=(x_1,x_2)\succsim (y_1,y_2)=y$ $\iff$ $x_1\geq y_1$ and $x_2\geq y_2$. 

1) You might like to argue whether this preference relation is strictly monotonic and continuous. 

2) Is the relation defined above complete?

Then, as a side dish, you might also reconsider your claim that continuity is the cause of transitivity.

Note: I just wrote this particular one with the purpose of providing a thought experiment. More in a way to challenge your understanding. I am not sure whether this example provides an answer to your question or not.