What is the difference between a probability and a probability density function?
$\bullet$ Is it true that "in a probability density function, the area under the curve tells you the probability"? So consider a Gaussian probability density function, $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$, now what is the probability of $x$ be equal to $\mu$? The area under a point in a curve is zero, isn't it?
$\bullet$ What does it mean when one says "the normal distribution is a probability distribution", so the total area under the normal curve is 1? Do we have "distributions" that are not a probability distribution?
Reference: Normal Distribution | Examples, Formulas, & Uses - Scribbr
I'm trying to understand this paper[Oded Regev. "On lattices, learning with errors, random linear codes, and cryptography", in Journal of ACM, 2009. Section 2. Preliminaries, pages 12-15].