If $g(x)$ is my density function, then $g(x_0)$ is simply $P(X=x_0)$?

Question

If $X$ is a continuous random variable with density $g(x)$, can I say that $g(x_0)$ is the probability that $X$ is equal to $x_0$

Answer 1

No. If $X$ is defined on $\mathbb{R}$, then $$ P(X = x_0) = \int_{x_0}^{x_0} g(x)~\mathrm{d}x = 0. $$ As $g$ is a probability density, the following holds: $$ \int_{-\infty}^\infty g(x)~\mathrm{d}x = 1 $$ So $g$ can not be $0$ everywhere, this means that $P(X = \cdot)$ and $g(\cdot)$ are never the same.