I'm taking a course in probability and statistics and I was given the following problem: Let $F:\mathbb{R} \to [0,1] $ be a continuous, strictly increasing function that satisfies $\lim_{x\to \infty}{F(x)}=1 \ \land \ \lim_{x\to-\infty}{F(x)}=0$. Prove that there exists a random variable $X$ that has F as a cumulative distribution function.
So, I can see that F satisfies all the conditions to be a CDF, but I can't assume that it's a differentiable function(which would allow me to define a probability density function). And if I can't prove the existence of a probability density function, then I can't prove the existence of a continuous random variable.
Any ideas?