Let the distribution function of continuous random variable X be f (x), and the density function be f (x), then why is p (x = x) = 0?

Let the distribution function of continuous random variable X be f (x), and the density function be f (x), then why is p (x = x) = 0?

According to the definition of distribution function, P {x = x} = f (x) - f (x-0) = 0