Q. What is P(X = x) for a continuous probability density function (pdf)?
In order to evaluate a pdf we use integration (unlike a pmf, which uses summation). When you integrate a function over a single value, you will always get zero.
\[\int_a^a f(x)dx = F(x)|_a^a = F(a) – F(a) = 0\]
This result is not useful. However, we can still integrate pdfs over a range of values to produce useful results. The pdf also tells us the shape of the distribution and is useful for finding cumulative density functions (CDFs)