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

I am reading a text that states that with a discrete random variable the probability mass function is used since x can take on particular values with a particular probability. However, it goes on to say, with a continuous case since the probability of a particular value of X is not measurable instead a PDF is used where a “probability mass” per unit per length around x can be measured. I think it’s the use of the word “mass” to describe a PDF is throwing me off. What are your thoughts?

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.