$P(X=c)=0$ for normally distributed $X$?

Question

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?

Answer 1

If $X$ is any continuous random variable (e.g. normally distributed), then $P(X=c) = 0$ for all $c$ in $\mathbb{R}$, no matter what distribution $X$ has or what point $c$ is. Points are too small to have non-zero probability of being hit by $X$.

If by the quantity $b$ you mean the variance -- normally, we write $X \sim N(\mu,\sigma^2)$ -- then $b$ is zero if and only if $X$ is constant. No normally distributed r.v. can have zero variance. If $X$ is constant with constant value $c$, then $P(X=c) = 1$ and $P(X=d) = 0$ for all other $d \ne c$.