Must probability density be continuous? From other materials that I've read, the probability density of a continuous random variable must itself be continuous. Is this correct? If it is, I don't understand why that would be so, why can't the probability change abruptly?
 A: Michael Chernick asks for an example of a probability distribution with a density that is everywhere discontinuous.
As discussed in this question, there exists a measurable set $A \subset \mathbb{R}$ such that for every interval $I$, we have $0 < m(A \cap I) < m(I)$, and moreover $m(A) < \infty$.  Then $f(x) = \frac{1}{m(A)} 1_A(x)$ is a nonnegative measurable function with $\int_\mathbb{R} f(x)\,dx = 1$, so it can be taken as the density of a continuous probability distribution.  $f$ is nowhere continuous because every interval contains points of $A$ and $A^C$.  Moreover, any function $g$ with $f=g$ a.e. is also nowhere continuous.
A: Take $f(x) = 2x$, $0\le x \le 1$, and 0 otherwise.  This is a density function which is not continuous.  
A: Although valid I don't think the triangular density given by ncmathsadist is a good example.  The U[0.1] density is discontinuous too because of the abrupt rise at x=0 and drop at x=1.But both these densities are continuous within their domain.  I think a better example  would be one with a discontinuity in its domain.
Consider the density f(x)=2x for 0<=x<=1/2
and f(x)=(5-2x)/16 for 1/2

Certainly a density can have many such discontinuities.  But can anyone give an example of a true probability distribution with a density that is everywhere discontinuous?
A: No, need not be. However, the cumulative distribution function (CDF), is always continuous (mayn't be differentiable though) for a continuous random variable.
For discrete random variables, CDF is discontinuous.

