Meaning of probability density function - continuous random variables

Question

Suppose we have a random variable X uniformly distributed over the interval (0,1). The probability density function of X is given by: $$f(x)=\left\{\begin{array}{l} 1 \space\space if \space\space a<x<b \\ 0 \space \space otherwise \end{array}\right.$$

So for example we have:

$f(0.125)=1$

$f(0.567987) = 1$

$f(0.7654) = 1$

....and so on.....

Now, my question is:
If $f(X=x_j)= 1 \space\space and \space x \in (0,1) $ (it's not the probability that $X=x_j$) what does this 1 represent? What is density function for a representation $x_j$ of a random variable X?

Answer 1

For a continuous probability, the pdf $f$, means that locally (instantaneously) probability is accumulating at a rate of $f(x)$ units of probability per unit of $x$.