You can make a parallel between $(a,b)$ and $(x,x+\delta x)$, with $\delta \rightarrow 0$. So, $\delta$ has the normal meaning of a tiny (very tiny) step in $x$, as in differential calculus. In the case of probability, one cannot have $p(x)$ with $x$ being a single value, because the domain of $x$ is infinite ($x$ is a continuous variable); and so $p(x)$ would be 0 for every $x$.
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