Why does MLE work for continuous distributions?

In the attachment below you can see the definition of the likelihood function. Likelihood

1) Whilst the explanation of why the whole max likelihood method is viable for discrete distributions is clear to me, I struggle to persuade myself why this approach is valid for continuous cases. Why is just multiplying the densities appropriate?

2) I don't seem to understand the explanation why the $\log(L(\theta))$ may be used either. In the same attachment point 1 declares that "Strictly monotonic increasing functions $g$ preserve order..." Whilst this is the case for $\log(L(\theta))$ I don't understand what exactly it justifies. If I understand correctly, $L(\theta)$ can be decreasing OR decreasing. So I don't see the connection.

At the same time another explanation (from another source) sounded: "This is because the two functions, $\log(L(\theta))$ and $L(\theta)$ are monotonically related to each other so the same MLE estimate is obtained by maximizing either one." In other words, no matter if $L(\theta)$ is growing or decreasing the $\log(L(\theta))$ will do the same. Would you agree that this is a more appropriate explanation?

• "just multiplying the densities" is a highly misleading phrase. Usually if one speaks of multiplying functions it means pointwise multiplication, i.e. the product of $x\mapsto f(x)$ and $x\mapsto g(x)$ is $x\mapsto f(x)g(x)$. You multiply two functions, each of just one variable, and what you get is still a function of just one variable. But that is not at all what is done with the density functions. You have $f_\theta(x_1)$ and $f_\theta(x_2)$, and those are functions of two different variables, and each on by itself is a function of just one variable, but the product is a function of two variables.
• It is true that typically $L(\theta)$ will sometimes increase and sometimes decrease as $\theta$ increases, and along with it $\log(L(\theta))$ sometimes increases and sometimes decreases as $\theta$ increases. But where is says "Strictly monotonic increasing functions $[\ldots]$ preserve order", the strictly monotonic function referred to is neither $L(\theta)$ nor $\log(L(\theta))$, but rather it is the logarithm function. The fact that $\log(L(\theta))$ and $L(\theta)$ are monotonically related is deduced from the fact that the logarithm function is strictly increasing. The two proposed explanations, of which you say one is more appropriate than the other, both say the same thing.