Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in \mathbb{R}^-$.
QUESTION: In general, for which (i) and/or (ii) am I allowed to push the limit inside and evaluate (i) $f(\lim_{b\to 0^+} \frac{a}{b})=f(-\infty)$, and (ii) $f(\lim_{b\to 0^-} \frac{a}{b} )= f(\infty)$?
I was told you can't do this for discontinuous functions (this function has jumps when limit approaches from left), which makes me concerned about doing this, even though the domain is the extended reals.
FYI: This $f$ is actually the cumulative distribution function of the standard normal probability density function (the standard normal, $\mathscr{N}(0,1)$ CDF). ((related to the erf(x) or even the complementary error function)).