TITLE: What does it mean to do MLE with a continuous variable
QUESTION [14 upvotes]: I am struggling with the semantics of continuous random variables. 
For example, we do maximum likelihood estimation, in which we try to find the parameter $\theta$ which, for some observed data $D$, maximizes the likelihood $P(\theta|D)$. 
But my understanding of this is $$P(\theta = x) = P(x\leq\theta\leq x) = \int_x^xp(t)dt = 0$$ so I am not sure how any $\theta$ can result in a non-zero probability.
Intuitively I understand what it means to find the "most probable" $\theta$, but I am having trouble uniting it with the formal definition.

EDIT: In my class we defined $L(\theta:D)=P(D|\theta)=\prod_i P(D_i|\theta)$ (assuming i.i.d, where $D_i$ are the observations). Then we want to find $\text{argmax}_\theta \prod_i P(D_i|\theta)$.
I was incorrect above about finding $P(\theta)$, but it seems to me we're still trying to find the maximal probability, where all probabilities are zero. Some answerers suggested that we're actually trying to find the max probability density but I don't understand why this is true.

REPLY [2 votes]: Here's still another way to view the MLE, that really helped clarify it for me:
You're taking the derivative of the pmf (With respect to whatever variable you're trying to isolate) and finding a local maximum by setting that derivative equal to 0.
That's what the MLE is.  To look at it from the viewpoint of a normal distribution, you're finding the exact value (Or the formula for it) of the peak (highest probability of occurring - the mean, in the case of the normal), because that's where its derivative changes direction (So it, for an instant, is 0 there)
The log step is because all taking a log does is flattens a curve (Such as you see here, in situations where it won't just flatten the curve you can't take the log) which doesn't change the local maximum at all, just lowers it -- but it almost always makes the derivative more straightforward.
Hope this helps!