Why is the probability that a continuous random variable equals a specific number zero? That is, why is P(X=b)=0 for any number b?

Answer 1

Prob(X=b) = Prob(b<=x<=b) <-- yes this is a true identity

= Prob(x<=b) - Prob(x<=b) <-- probability of between = probability of larger - prob. of smaller

= 0