The likelihood over a continuous random variable $X$ at $x$ with parameter $\theta$ is given by
$$L(\theta ; X=x) =f_X(x;\theta)$$ where $f_X$ is a pdf on $X$.
For calculating likelihood
In case of discrete random variable we use pmf in place of pdf. But in case of continuous we are using pdf instead of probability measure.
I am thinking of following reasons
1) PDF indirectly gives the measure of probability at particular assignment of random variable.
2) The probability over continuous random variable is 0 at a particular assignment and exists only in ranges.
Do my reasons make sense? else, are there any other reasons for using pdf?
Answer 1) yes correct, but there is no problem because in practice you work with samples where you have realizations/observations of the variables, for which you want to measure the likelihood. In that sense we talk about the realizations of the random variables.
Answer 2) Correct again, but you will not calculate the probability of infinitesimal differential of the variable: in practice you work with the realizations of the variables that are for example real values with a certain discrete approximation describing the observations for that variable. The continuous function will ensure that, for every real realization, you will have a certain probability assigned to it. That’s the meaning of continuous. You will not use the probability of an infinitesimal variation of the variables when it comes to the likelihood in real practice: in the latter case of differentials you will use an integral between two defined bounds anyway.
So, if I got your points, they are correct theoretically, but consider that in practice you work with likelihood functions in order to measure the probability of a certain set of realizations of the random variable or stochastic process.