I am reading how $f(x)dx$ is an approximation for $\mathbb{P}[x \leq X \leq x+d x]$ where $X$ is a continuous random variable and $f$ is it's density function, and was wondering why this is precisely. Here is the passage I was reading:
To connect density $f(x)$ with probabilities, we need to look at a very small interval $[x, x+d x]$ close to $x$; then we have $$ \mathbb{P}[x \leq X \leq x+d x]=\int_x^{x+d x} f(z) d z \approx f(x) d x $$
I was wondering why $f(x)dx$ is a good approximation- if $f(x)$ is constant about $x$ then this seems like a great approximation, but what if f curves sharply up over an infinitesimal change in $x$? Why would this not invalidate the soundness of $f(x)dx$ as an approximation for this?
$f(x)$ is not the approximation for $P(x\leq X \leq x+dx)$. It is the product $f(x)dx .$