Problem with median and even functions

Question

f(x) is even, so the median will be always 0? Can I assume this?

Assume that a continuous random variable $X$ has a probability density function $f$ satisfying $f (x) = f (-x)$ for all $x \in \mathbb{R}$. If $M$ is the median of $X$, then: What do I choose as a correct answer?

1) $M = 1$.

2) $M = 0$.

3) Nothing can be concluded about $M$.

Answer 1

$0$ is a median since you will have $\mathbb P(X \le 0) \ge \frac12$ and $\mathbb P(X \ge 0) \ge \frac12$

But consider $f(x)=\frac12$ when $-4 \lt x \lt -3$ and $3 \lt x \lt 4$ while $f(x)=0$ otherwise. This will have $-3$ and $3$ also as medians and any values inbetween