By [definition](https://en.wikipedia.org/wiki/Median#Probability_distributions), $m$ is a median of the random variable $X$ if and only if $$\mathrm{Pr}[X ≤ m] ≥ \tfrac12 \text{ and } \mathrm{Pr}[X ≥ m] ≥ \tfrac12.$$

(Note that, in general, $X$ may have multiple medians if there is an interval of values satisfying the definition, and that it's possible for either or both probabilities to be strictly greater than $\tfrac12$ if $\mathrm{Pr}[X = m] > 0$.)

Also by [definition](https://en.wikipedia.org/wiki/Monotonic_function), a function $g: \mathbb R \to \mathbb R$ is monotone increasing if and only if $$a ≤ b \implies g(a) ≤ g(b)$$ for all $a,b \in \mathbb R$.

(A function $g$ is called _strictly_ monotone increasing if $a < b \implies g(a) < g(b)$, but we don't actually need this stronger property here.)

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Now, let $m$ be a median of $X$ and let $g$ be monotone increasing.  Then $X ≤ m \implies g(X) ≤ g(m)$ (and $X ≥ m \implies g(X) ≥ g(m)$), and thus $$\begin{aligned}
\mathrm{Pr}[g(X) ≤ g(m)] ≥ \mathrm{Pr}[X ≤ m] &≥ \tfrac12 \text{ and } \\
\mathrm{Pr}[g(X) ≥ g(m)] ≥ \mathrm{Pr}[X ≥ m] &≥ \tfrac12.
\end{aligned}$$

In other words, if $m$ is a median of $X$ and $g$ is monotone increasing, then $g(m)$ is a median of $g(X)$.

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**Ps.** Note that the result above holds for any real-valued random variable $X$ and any monotone increasing function $g$, even if the medians of $X$ and/or $g(X)$ are not uniquely defined.

However, it only guarantees that $g(m)$ is _some_ median of $g(X)$, but not that any particular rule for picking a "canonical" median $m$ for $X$ — such always taking the low end, high end or the midpoint of the interval of possible medians — will necessarily yield $g(m)$ when applied to the distribution of $g(X)$.

Also, if the function $g$ is discontinuous, $g(X)$ may have multiple medians even if $X$ does not, while if $g$ is not strictly increasing, $X$ may have multiple medians even if $g(X)$ does not.  And if $g$ is not strictly increasing, then it's also possible for $g(m)$ to be a median of $g(X)$ even if $m$ is _not_ a median of $X$.

In particular, even if we define $\operatorname{median}[X]$ to be the set of _all_ medians of $X$, the most we can say in general is that $g(\operatorname{median}[X]) \subseteq \operatorname{median}[g(X)]$.

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**Pps.** The result above can also be straightforwardly generalized to arbitrary [quantiles](https://en.wikipedia.org/wiki/Quantile) by defining $m$ to be a $p$-quantile of $X$ (for some $p \in [0,1]$) if and only if $$\mathrm{Pr}[X ≤ m] ≥ p \text{ and } \mathrm{Pr}[X ≥ m] ≥ 1-p$$ and using essentially the same proof to show that, if $g$ is monotone increasing, then $g(m)$ is a $p$-quantile of $g(X)$.