A continuous function defined on an interval can have a mean value. What about a median? A function can have an average value
$$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$
Can a continuous function have a median?
How would that be computed?

Sure a continuous function can have a median, as for if it always has a median I am unsure of. 
Let's suppose we have a some function $f$ and it is continuous on $[a,b]$. We want to find its median $m$. We know that if $m$ is a median than if we take any value $c \in [a,b] $ at random then the $P(c \leq m) = $$P(c \geq m) = 1/2 $. Now if we want to construct such a function we need to "normalize" it in a sense. 
So let's say we take the integral of $f$ over $[a,b]$ and it is equal to $S$. 
$\int_a^b f(x)dx = S$
Now let $=g(x) = \frac{1}{S}f(x)$ so that we have
$\int_a^bg(x)dx = 1$. 
Now if we want to find the median of $g(x)$ and therefore $f(x)$ we simply need to find an $m$ such that 
$\int_a^mg(x)dx = 1/2$ 
