# Is zero random variable a continuous random variable?

In my probability textbook I got stuck on the following problem: cosntruct a positive (it means $\geq 0$) real function $f(x)$ (not necessarily continuous, of course) such that it is constantly $0$ for every $x \leq 0$ and $$1 = \int_{- \infty}^{x} f(t) \ \! \mathrm{d}t$$ for all $x >0.$ This problem should be obvious (according to the authors), yet I am not sure it is even true. Any suggestions? Thanks a lot.

The conditions you specify give the cumulative density function $F(x) = \mathbb{1}_{[0,\infty)}(x)$ which is indeed a valid cumulative density function ($\lim_{x\to -\infty} F(x) = 0, \lim_{x \to \infty} F(x) = 1, \lim_{x \to a^+} F(x)= F(a)$).
This distribution is a point mass at $0$, which as the other answer points out has a "density" as the Dirac delta distribution.
• Let me suggest to stop being shy and to squarely state that there is no such function $f$. This is the most important message to bring to the OP. – Did Oct 12 '14 at 14:46