Interpretation of probability density function - uniform distribution

#1
Dear all,

I'm having some problems in the interpretation of the probability density function. Until now I thought that f(x) for a continuous random variable was equal to the probability that X took a specific value in a tiny interval dx. But this interpretation is obviously wrong because in the case of the uniform distribution between 0 and 1, f(x) = 1, which according to my first interpretation is not possible. So how do we have to interpret specifically the probability density function?

Thank you
 
#2
Intuitively, the probability of X falling in the tiny interval between x and x+dx is f(x).dx. If f(x) = 1 then the probability of X falling between x and x+dx is 1.dx, which makes sense.