Draw random numbers from an incomplete uniform distribution

Goal

I am trying to build a function that allows me to draw a specific number of random numbers from an an incomplete uniform distribution.

What do I call an incomplete uniform distribution?

I call an incomplete uniform distribution a probability distribution where each value of X within a series of boundaries have an equal probability to be picked. In other words it is a uniform distribution with holes as represented below

x = list(12:25, 34:54, 67:90, 93:115)
y = 1/sum(25-12, 54-34, 90-67, 115-93)
plot(y=rep(y, length(unlist(x))), x=unlist(x), type="n", ylab="Probability", xlab="X")
for (xi in x)
{
    points(xi,rep(y, length(xi)), type="l", lwd=4)
}

Ugly Solution

Here is a slow and ugly solution

IncompleteUnif = function(n,b)
{
    #################
    # "n" is the desired number of random numbers
    # "b" is a list describing the boundaries within which a random number can possibly be drawn.
    #################
    r = c() # Series of random numbers to return
    for (ni in n)
    {
        while (length(r) < n) # loop will continue until we have the "n" random numbers we need
        {
            ub = unlist(b)
            x = runif(1,min(ub), max(ub)) # one random number taken over the whole range
            for (bi in b) # The following loop test if the random number is withinn the ranges specified by "b"
            {
                if (min(bi) < x & max(bi) > x) # if found in one range then just add "x" to "r" and break
                {
                    r = append(r,x)
                    break
                }
            }
        }
    }
    return (r)
}

b = list(c(5,94),c(100,198),c(220,292), c(300,350))
set.seed(12)
IncompleteUnif(10,b)
[1]  28.929516 287.132444 330.204498  63.425103  16.693990  66.680826 226.374551  12.892821   7.872065 140.480533