Intuition of pdf of a continuous random variable What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if you plug a single value into $f(X=x)$, it outputs a value. What is the intuitive understanding of this value for a given x? 
For example, in the above image, the value of the red curve at x=0 is 0.4. How can 0.4 be interpreted?
 A: The value of a continuous density function at a given point is also called its likelihood. Yes, the probability of getting any specific number in the range is exactly equal to 0—there are uncountably many options. However, it remains true that observations may cluster, and some values are still more likely than others. 
For example, the probability that a standard normal variable $Z$ takes on the value $0$ is exactly the same that it takes on the value $1\times 10^8$, namely 0. But you will see $0$s much more often. This is reflected in the value of the pdf at the point, as any interval containing 0 is going to have a larger area, and thus larger probability, than any equal-length interval not containing 0 (the normal is symmetric and unimodal at 0).
So, informally speaking, the value of 0.4 is not useful for its magnitude in and of itself (although it is very useful in maximum and relative likelihood analyis), but it can be compared with other values on the curve to see which values may be more likely. But, as said earlier, probability only has positive value for intervals of continuous distributions.
A: It is a density value.  Density only has meaning when it is collectively considered over an area or volume.
A: In the case of a discrete function it may output a value. A continuous function will output zero for a particular point since it represents an infinitesimal. Here is a good article explaining it from ground up:
The idea of a probability density function
Hope that helps.
