probability density function In a book the following sentence is told about probability density function at point $a$: "it is a measure of how likely it is that the random variable will be near $a$." What is the meaning of this ?
 A: The meaning is that if the probability density function is continuous at $a$, 
then the probability that the random variable $X$ takes on values in a short interval of length $\Delta$ that contains $a$ is approximately $f_X(a)\Delta$. Thus, for example,
$$P\left\{a - \frac{\Delta}{2} < X < a + \frac{\Delta}{2}\right\} \approx f_X(a)\Delta$$
and the approximation improves as $\Delta \to 0$. In other words,
$$\lim_{\Delta \to 0}\frac{P\left\{a - \frac{\Delta}{2} 
< X < a + \frac{\Delta}{2}\right\}}{\Delta} = f_X(a)$$
at all points $a$ where $f_X(\cdot)$ is continuous.

"it is a measure of how likely it is that the random variable will be near $a$."

If by "near $a$" we mean in the interval $\left(a-\frac{\Delta}{2}, a-\frac{\Delta}{2}\right)$ where $\Delta$ is some fixed small positive number,
then, assuming that $f_X(\cdot)$ is continuous at $a$ and $b$, the
probabilities that  $X$ is near $a$ and near $b$ are respectively
proportional to $f_X(a)$ and $f_X(b)$ respectively, and so the values
of $f_X(\cdot)$ at $a$ and $b$ respectively can be used to compare
these probabilities.  $f_X(a)$ and $f_X(b)$ respectively are a measure 
of how likely $X$ is to be near $a$ and $b$.
A: It is not a meaningful sentence.
For a continuous random variable, the value of the probability density function (pdf) at a point tells us nothing about the probability that the random variable lies near that point. This is because the random variable is continuous, so the probability that it evaluates to any given point is of measure zero.
What is meaningful is to talk about the integral of the pdf near the point $a$, as this yields a probability. So a much better sentence would be:
"The definite integral of the probability density function over a small neighbourhood of the point $a$ is a measure of how likely it is that the random variable will be near $a$"
A: It would be illustrative to compare the case with a discrete R.V. With a discrete R.V, the pdf gives the distribution of the probabilities at specific points i.e, it gives you the probability that the R.V takes a particular value. With a continuous R.V, the pdf gives the distribution of probabilities over an interval i.e, it gives you the probability that the R.V takes values in an interval.
Mathematically, if $f(x)$ is the pdf of a discrete R.V - X, then $f(x) = P(X = x)$. If $f(x)$ is the pdf of a continuous R.V - X, then $f(x) = P(X \le x)$.
If $f(x)$ is the probability density function of a continuous R.V, then, $f(x)dx$ is the linear approximation to the cumulative probability of $X$ over the interval $[x - dx, x]$. (If you integrate the pdf, then as Colin Bowers said earlier, you get the cumulative distribution function, which gives the exact cumulative probability over any arbitrary interval $[x - dx, x]$.
It must also be noted that the 'differential probability' - $f(x)dx$ also depends on the interval $[x - dx, x]$ and not just $f(x)$. How small the interval should be, depends on how rapidly f(x) fluctuates.If $f(x)$ varies very slowly, you can take a larger neighbourhood, while if it varies rapidly, you would have to consider a smaller neighbourhood(for the linear approximation to be reasonably accurate).
A: There are many version of it. In the textbook I used, it says a probability density function$(f(x))$ is just a nonnegative function whose integral from $-\infty$ to $\infty$ is $1$.
What means more is the "Cumulative Distribution Function"$(F(x))$, which is defined as
$$F(x)=\int_{-\infty}^xf(t)dt$$
This shows the probability of $X\leq x$.
The probability of $X=x$ does not make much sense in continuous case, but as your book says, the probability of $X$ "near" $x$ can be "sort of" described by $f(x)$, as $$P(x_1\leq X\leq x_2)=\int_{x_1}^{x_2}f(t)dt$$
So, by looking at the graph of $f(x)$, you can get a rough idea about how $F(x)$ will look like and get a sense of probability "near" $X=a$.
You can think of the discrete case as simple function of continuous case.
