Would it be accurate to describe a PDF as a probability mass per unit length

A PDF $f$ for a continuous random variable $X$ satisfies $P(a \le X \le a+\delta) = \int_a^{a+\delta} f(x) \, dx$ for $\delta > 0$.

If $\delta$ is small, then the integral is $\approx \delta f(a)$ (the area under the graph of $f$ on the interval $[a, a+\delta]$ is approximately the area of the rectangle with width $\delta$ and height $f(a)$). Rearranging yields $$f(a) \approx \frac{P(a \le X \le a+\delta)}{\delta}.$$

The fraction on the right-hand side is the "probability mass per unit length around $a$." I would think of "probability mass" of an event simply as the probability of the event.