Could someone explain why this probability ISN'T 0? Despite being a CDF?

$P(X = 1/2) \neq 0$? Shouldn't it be $P(X = 1/2) = F(1/2) - F(1/2) = 0$
Could someone explain why this probability ISN'T 0? Despite being a CDF?

$P(X = 1/2) \neq 0$? Shouldn't it be $P(X = 1/2) = F(1/2) - F(1/2) = 0$